Introductory thoughts
What does it mean to treat the Earth as an inertial frame? I would say it means that we pretend the Earth is an infinite plane, moving with a constant velocity in some direction. This accounts for many everyday phenomena, since we are so small we essentially always live in the tangent plane of the Earth, and for those of us who aren't professionals in aerospace, objects we throw in the air tend to come down before the tangent plane of the Earth changes by very much.
This assumption is good so long as the deviation of some motion of interest, is smaller than some threshold of acceptability, from the motion accounting for non-inertial effects. Note that the deviation of the motion will essentially always rise above any threshold given enough time (even if I apply $10^{-12}$ N to you, if I do it continuously over a long enough time you will eventually notice you are shifted relative to where you would be if no force is applied. The buildup of a small effect that always points in the same direction until it becomes a large effect, are sometimes known as secular variation.
We can illustrate these general points with the simple example of uniform acceleration in 1 dimension. Then the trajectory of an "inertial observer" is $x_I (t) = x_0 + v_0 t$, and an accelerated observer which "matches" the inertial observer at $t=0$ has a trajectory $x_A (t) = x_0 + v_0 t + \frac{1}{2} a t^2 = x_I(t) + \frac{1}{2} a t^2$. We're interested in the error we make in ignoring the acceleration term, compared to some acceptable level of error in the position $\Delta x$
\begin{equation}
\epsilon = \frac{\left| x_I(t) - x_A(t) \right| }{\Delta x} = \frac{a t^2}{2\Delta x} = \left(\frac{t}{t_A} \right)^2
\end{equation}
where $t_A = \sqrt{2 \Delta x / a}$ is the time scale over which the non-inertial effects become important.
The error grows with $t$, but only becomes large when $t$ is of order the timescale $t_A$. Given that we have some fixed tolerance in position $\Delta x$, and that a typical experiment happens on some human timescale $t_H$, we can neglect the effect of acceleration if $t_H \ll t_A$, which requires $a$ to be smaller than $\Delta x / t_H^2$. This form of this scaling argument gives us intuition for what to expect in more general cases.
Rotation of the Earth
Probably the most obvious non-gravitational, non-inertial effect is the rotation of the Earth. I think this point was very well covered by other answers, and I agree the conclusion that the magnitude of things like the Coriolis force are so small that you don't notice them over "human" timescales. The parameter that controls the smallness of rotational effects of the Earth is the angular velocity.
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Well, actually that's not quite true since the Earth's motion is very complicated, and in addition to the daily rotation there is also precession and nutation -- while any non-inertial forces from these effects are miniscule by any reasonable defintion, understanding secular effects from precession are crucial for astronomers).
Gravitational and tidal effects from the Earth and moon
Gravity makes this a bit tricky, but I think a uniform gravitational field can be considered a "benign" form of non-inertial-ness, since by the equivalence principle a uniform gravitational field is simply a uniformly accelerating frame of reference. So I think we can say the Earth is "inertial enough," so long as the gravitational field is "close enough" to uniform. The gravitational acceleration $\vec{g}$ for an isolated, spherical body of mass $M$ is
\begin{equation}
\vec{g} = \frac{GM}{r^2} \hat{e}_r = \frac{GM}{R^2} \left(1 - 2 \frac{\delta r}{R} + \cdots \right) \vec{e}_r
\end{equation}
where $\hat{e}_r$ is a unit vector pointing from the observer to the center of the Earth, and $R$ is the radius of the Earth. In the second equality, we have written the distance from the Earth to observer as $r=R+\delta r$, where $\delta r$ is the radial distance of the point to the surface of the Earth, and expanded in $\delta r / R$. Tidal forces due to a non-uniformity of Earth's acceleration come from the second term, $\delta r/R$. So long as the height of an object above the Earth's surface is a small fraction of the Earth's radius, we can treat the Earth's gravitational field as uniform. This is a good approximation if you throw an apple 1 meter in the air. It is not a good approximation if you launch a satellite into orbit.
You can try to account for more subtle things like the fact that the Earth is not a perfect sphere. The next best approximation would be to say the Earth is an ellipse; you will get another set of corrections proportional to the small ellipticity of the Earth, $e$. Even more subtle effects come from the local topography of the Earth; these will be suppressed by something like the height of the topography variation (eg the height of a mountain) over the radius of the Earth. All of these effects are in fact noticeable if you perform high precision measurements of the Earth's gravitational field.
We can also look at tidal forces from the moon. Here $\delta R$ is the same, but now $M$ is smaller (it is the mass of the moon instead of the Earth) and $R$ is much bigger (the difference of the moon's center to the Earth's surface), so the tidal force $\sim GM \delta r / R^3$ is much, much smaller than the tidal force due to the Earth itself. We do notice the tides of course; this basically boils down to the facts that we are small compared to the Earth so we do notice small changes, and 12 hours is a substantial fraction of the period of the Earth's rotation so $\omega t$ becomes of order 1. More precisely, what distinguishes the moon's tidal force from the Earth's own tidal force is that $R$ (as I've defined it) is time dependent, and it's often much easier to see a time-varying small effect than a constant one.
Motion through the cosmos
Finally, considering motion of the Earth through the cosmos, as was also discussed in other answers, freely falling motion is inertial. If there are genuinely non-gravitational forces that the Earth / solar system / galaxy experience, these must act on such a long time scale that they will not be significant for Earth-based measurements.
Wrapping up
I think most of the points here would generalize to other planets, or at least gives you a condition for checking if it would generalize. While you need to reason about each effect on a case-by-case basis, there is always (a) some small parameter controlling the non-inertial effect, be it the rotational velocity for Earth's rotation, $\delta R/R$ for tidal effects, the ellipticity, etc, and (b) some time scale that tells us when secular effects become important.