Why don't we feel the subtle speed change of Earth's elliptical orbit? Earth's orbit is a slight ellipse, so to conserve momentum its speed increases when it is closest to the Sun. If the speed changes there is an acceleration. If there is an acceleration there is a force. Even if the change is small and gradual, wouldn't we experience a force because the Earth is so massive?
 A: We don't feel any acceleration because the Earth and all of us humans on it is in free fall around the Sun. We don't feel the centripetal acceleration any more than the astronauts on the ISS feel the acceleration of the ISS towards the Earth.
This happens because of the way general relativity describes motion in gravitational field. The motion of a freely falling object is along a line called a geodesic, which is basically the equivalent of a straight line in curved spacetime. And because the freely falling object is moving in a straight line it experiences no force.
To be a bit more precise about this, the trajectory followed by a freely falling object is given by the geodesic equation:
$$ \frac{\mathrm d^2x^\alpha}{\mathrm d\tau^2} = -\Gamma^\alpha_{\,\,\mu\nu}U^\mu U^\nu \tag{1} $$
Explaining what this means is a bit involved, but actually we don't need the details. All we need to know is that the four-acceleration of a body $\mathbf A$ is given by another equation:
$$ A^\alpha = \frac{\mathrm d^2x^\alpha}{\mathrm d\tau^2} + \Gamma^\alpha_{\,\,\mu\nu}U^\mu U^\nu \tag{2} $$
But if use equation (1) to substitute for $d^2x^\alpha/d\tau^2$ in equation (2) we get:
$$ A^\alpha = -\Gamma^\alpha_{\,\,\mu\nu}U^\mu U^\nu + \Gamma^\alpha_{\,\,\mu\nu}U^\mu U^\nu = 0 $$
So for any freely falling body the four acceleration is automatically zero. The acceleration you feel, the "g force", is the size of the four-acceleration - technically the norm of the four-acceleration or the proper acceleration.
Nothing in this argument has referred to the shape of the orbit. Whether the orbit is hyperbolic, parabolic, elliptical or circular the same conclusion applies. The orbitting observer experiences no acceleration.
You might be interested to read my answer to How can you accelerate without moving?, where I discuss this in a bit more detail. For an even more technical approach see How does "curved space" explain gravitational attraction?.
A: John Rennie's answer is right from the viewpoint of General Relativity -- but since the question is tagged with Newtonian mechanics, it deserves a Newtonian answer too.
In the Newtonian framework, I think the best answer to "why don't we experience this force" is that we can't feel forces that apply to our body at all. What we actually experience with our senses are only forces between different parts of our body.
When you stand on the surface of Earth, you feel neither the gravitational pull of the Sun nor the gravitational pull of the Earth. Strictly speaking you don't even feel the contact force between your footsoles and the ground -- but you do feel the compressive force between the skin of your feet and the bones inside the foot. And to a lesser extent you feel your bones being compressed, and your flesh being stretched by hanging from your skeleton. All these internal forces balance out the gravitational pull on your body, such that there's zero net force applying to each part of it (ignoring the pull from the sun and moon), and you stay in place compared to the earth.
This is what produces the sensation of being pulled towards the earth: the internal forces in your body that resist that pull.
However, for the pull from the sun, there's nothing that balances it out. Every particle in your body simply falls towards the sun, with the acceleration given by the strength of the sun's gravitational field -- and every particle in the earth and in the air around you is doing the same, so no internal forces are needed anywhere to keep the various parts of your body in the same relative position. Therefore there is nothing to feel.
A: John Rennie has answered the question in terms on general relativity, but it can also be answered with Newtonian physics. Your question is very similar to this one: 
Why does the moon stay with the Earth?
and I can refer you to my answer there. In short, the Sun isn't only pulling on the Earth itself, it's pulling on everything on it as well, including us, with the same gravitational force. Therefore we experience the same gravitational acceleration due to the Sun as the rest of the Earth does. From Galileo's and Newton's laws of motion, it follows that we move on the same free-fall path as the Earth around the Sun, so we remain stationary relative to the Earth.
A: Even if the orbit were a perfect circle, there's some acceleration towards the sun. If there weren't acceleration then the earth would move in a straight line (instead of a circle); but it doesn't move in a straight line therefore there's acceleration.
In a sense, the earth doesn't feel the acceleration because it doesn't try to resist it: if you stand on something then you resist gravity (resist falling) and feel a force on your feet; if you don't stand on something and fall then (ignoring air resistance) you feel nothing (except maybe nausea because you're accustomed to feeling gravity).
An orbit can be described as a situation where instead of "falling onto" something you perpetually "fall around" it. Because you're in an infinite "free fall" (whether circular or elliptical) you feel no force -- there is a force (of gravity) but you don't resist it (you don't push against it) and so you don't feel it.
A: You definitely don't need to use General Relativity to answer this question.
It depends upon what you mean by "feel". If "feel" means "detectable by sophisticated instruments" then, yes, it can be "felt". But your body is not a very sophisticated detection instrument.
According to what I've read elsewhere, the Earth speeds up by $1000$ $m/s$ as it moves from its greatest distance from the Sun to its closest distance to the Sun. That takes six months or, roughly, 15,768,000 seconds. Use the following equation to roughly figure the acceleration:
$$V_{f} - V{o} = a*t$$
The acceleration of the Earth comes out to about $0.0000634$ $m/s^2$.
The whole Earth and everything on it is roughly accelerating at that rate and your body can't detect that very minor acceleration.
A: According to the Equivalence Principle a free falling system cannot locally detect a gravitational field. However Earth is a large enough system such that non-local effects turn out to be appreciable. Solar tides are - although small - detectable. So in principle one can experience the Sun's gravitational field even though we are in free fall. What I claim is that the acceleration change through the elliptical orbit is too small.
The Earth's angular momentum, with respect to the focus of the ellipse, is $L=mr^2\dot\phi$, where $m$, $r$ and $\dot\phi$ are the mass, the distance to the center of the Sun and the angular velocity, respectively. Therefore
$$mr_p^2\dot\phi_p=mr_a^2\dot\phi_a,$$
where the indices $p$ and $a$ denotes "perihelion" and "aphelion". For an ellipse,
$$r_p=\frac{r_0}{1+\epsilon},\quad r_a=\frac{r_0}{1-\epsilon}.$$
Hence
$$\frac{\dot\phi_p}{\dot\phi_a}=\left(\frac{1+\epsilon}{1-\epsilon}\right)\approx 1,0340,$$
since the Earth's orbit eccentricity is, $\epsilon\approx 0,0167$. We have a change of about three per cent in six months. The mean angular acceleration is 
$$\bar\alpha=\frac{0.0340\cdot\phi_a}{180\cdot 24\cdot 60\cdot 60}\sim 10^{-9}\phi_a\, \mathrm{rad/s^2}.$$
Notice that $\phi_a$ is of order 
$$\frac{2\pi}{365\cdot 24\cdot 60\cdot 60}\sim 10^{-7}\, \mathrm{rad/s}.$$
So the angular acceleration is of order $10^{-16}\mathrm{rad/s^2}$. If you multiply this value by the mean distance to the Sun, $r\sim 10^{11}\, \mathrm{m}$, we get an acceleration of order $10^{-6}\, \mathrm{m/s^2}$. That is negligible compared to the acceleration due to the Earth's gravity, $9.8\, \mathrm{m/s^2}$.
A: My answer is more metaphysics than physics.
The reason we do not "feel" the acceleration is that the change is within the tolerances of our bodies. That said, I am sure there have been people born who are more attuned to these forces. But for the most part, for most of use, there are so many forces acting on our senses that the acceleration of the earth is one we have learned to ignore, or just cannot feel.
An example is a seismograph. A simple one can be made from a pencil paper, and a flexible piece of plastic. With greater tremors this would work fine to mark size of earthquakes. However, the more rigid the plastic, the less movement you will see, and the greater the force applied to it required to make it move. Professional seismographs are made of much more sensitive material.
We are like that rigid plastic. We feel changes, but we are not made for detecting subtle changes such as the earth's acceleration.
That is not to say we cannot develop the skill to be more attuned to the earth's speed. In my martial arts studies, I have seen amazing "superhuman" feats that most anyone can do, if only one would spend the required time to develop such skills.
