Do planets experience time dilation as they orbit the sun and if so what effect would this have on their orbit Do the planets experience time dilation as they increase / decrease speed around the sun and are experiencing higher and lower gravitational forces as they move from a closer position i.e. the perihelion and a further position, the aphelion. If time dilation is present would it affect the orbit of the planet however small ?
 A: A satellite orbiting some large mass experiences time dilation from two sources:


*

*the gravitational time dilation due to the gravitational field of the central mass

*the special relativistic time dilation due to its speed
If you're interested I show how to calculate the combined effect of these two sources in my answer to Is gravitational time dilation different from other forms of time dilation? but for now let's just quote the result. For an observer on the planet, and compared to an observer far from the star, time runs slow by a factor of:
$$ \frac{d\tau}{dt} = \sqrt{1-\frac{3GM}{c^2r}} $$
So for example for a satellite orbiting the Sun at the Earth-Sun distance we'd put $M$ in as the mass of the Sun and $r$ as the Earth-Sun distance, and this gives us:
$$ \frac{d\tau}{dt} = 0.999999985 $$
This works out to be about half a second a year i.e. a clock at the Earth-Sun distance loses about half a second a year compared to a clock far from the Sun.
A sidenote: I've talked about a clock at the Earth-Sun distance rather than a clock on the Earth because a clock on the Earth runs slower than this. On the Earth the Earth's gravitational field causes an additional time dilation in addition to the time dilation caused by the Sun.
Finally, you ask:

If time dilation is present would it affect the orbit of the planet however small?

and the answer is yes but it's more complicated than that. Time dilation is just one aspect of the differences between relativity and Newtonian mechanics. We tend to talk about it because it's the most easily measurable effect. To calculate how relativistic effects modify the orbit of a planet is rather complicated. 
It has no effect on the orbit if the orbit is circular, but all the planets in the Solar system have orbits that are elliptical i.e. on one side of the orbit (the perihelion) they are closer to the Sun than on the other side of the orbit (the aphelion). This means at perihelion the relativistic effects (including time dilation) are greater than at aphelion, and this does affect the orbit. In fact it is what causes the anomalous perihelion advance of Mercury, so it has an easily observable effect. All the planets have a similar anomalous perihelion advance but the effect get smaller very quickly as you move away from the Sun so it is only easily measurable for Mercury.
A: $\let\D=\Delta$
I would object mainly to a word used in the title and also repeated by
@John Rennie: "experience". My english is not very good, but I believe
this verb brings with it a subjective flavour, something one can feel
or experiment, observe himself. Nothing could be farther from what
actually happens. 
People living on a planet have no sensation of time dilation. They
live as usual, can make physics experiments which give the same
results as in any other place in the Universe. Only by exchanging
signals with outside world they could see something strange, but in
different ways for the two kinds of dilation.
As to SR dilation what they would "see" is that time marked by a
faraway "stationary" clock is dilated wrt theirs. Instead they would
observe a real gravitational dilation of their own clocks. But this is meaningless if we don't give an exact definition of terms. Let's consider John's equation
$$d\tau/dt = 0.999999985.\tag1$$
The exact meaning is the following. The orbiting satellite brings a
clock, maybe an atomic clock to be sure of its stability and rate.
This is $\tau$. We have also to assume that wherever in space there are
"stationary" clocks, marking Schwarzschild time. And this is much
more difficult to explain.
First, what do I mean by "stationary"? I mean wrt to an inertial
nonrotating frame. This can be ensured by astronomical observations
and exchange of signals with faraway spaceships. Second comes
Schwarzschild time $t$. It's the time of faraway stationary clocks, where
"faraway" means that Sun's gravitational influence may be neglected.
But don't believe we could simply rely on a lot of cesium clocks to
define Schwarzschild time in Sun's neighbourhood. It is not so,
because those clocks would suffer GR gravitational delay. We can use
cesium clocks for frequency stability, but must set their running
(maybe with an elctronic contraption). In practice, assume a faraway
master clock is sending signals - say once per second. All clocks, even
within solar system, will have their running to be set so that they tell to
be receiving signals once per second as well.
Note that this sets clocks running, i.e. ensures they are neither fast
nor slow, but says nothing as to their being forward or back. This too can be done, i.e. all Schwarzschild clocks can be synchronized with the master, but I have to shorten my post...

Now we are ready to understand the meaning of eq. (1). $\tau$ and $t$
have been defined, but that's not all. Think of some phenomenon taking
place on our planet. It could be a simple thing like the falling of a
stone, or something more sophisticated. It will have a beginning and
an end. According local clock they will happen at times $\tau_1$ and
$\tau_2$. Define
$$\D \tau = \tau_2 - \tau_1.$$
The same phenomenon can also be recorded by those Schwarzschild
clocks which happen to be immediately near the planet's clock at
beginning and end (they will be different clocks, as planet is moving).
Let $t_1$ the time recorded by clock at beginning, $t_2$ the one
recorded by the other clock at end, and put
$$\D t = t_2 - t_1.$$
Then we are in a position to compute $\D\tau/\D t$, and verify if eq.
(1) holds true:
$$\D\tau/\D t = 0.999999985.$$
It holds, but I would by no means say that for an observer on the
planet time runs slow.
