Can only an object with mass feel forces?

If the pulley is massless then $I=0$ and so $T_1=T_2$ - in other words the tension is the same on each side of the pulley. This makes sense because there should be no net torque on a massless object (otherwise we would get an infinite angular acceleration, which is unrealistic).

So if $I=0$ we have

$W_1-W_2 = (M_1+M_2)a displaystyle Rightarrow a = frac {W_1-W_2}{M_1+M_2} = frac {M_1-M_2}{M_1+M_2}g$

The phrase "a massless rope" or "a massless pulley" in introductory mechanics classes should really be understood to mean that the mass of the object is small, but non-zero. We can do this by taking the limit as $M_p to 0$, which is effectively a sequence of cases where $M_p$ gets smaller and smaller and smaller but is never quite zero, and seeing what the behavior of the system is.

In the limit $M_p to 0$, everything is perfectly well-defined. In particular, the angular acceleration of the wheel is, for general $M_p$, $$ alpha = frac{(M_1 - M_2)g}{R(M_1 + M_2 + M_p/2)} $$ and in the limit as $M_p to 0$, this is a finite number. You will also find that in this limit, $T_1 = T_2$, so there is no net torque on the wheel; and since the moment of inertia of the wheel is also undefined, the equation $sum tau = I alpha$ becomes $0 = 0$, which is automatically satisfied.

In practice, saying that an object is "massless" in a mechanics problem is really just saying that its mass is so small that it can be ignored. Functionally, this is equivalent to saying that the object has no net force and no net torque on it. This is because $sum F = ma$, and if $m approx 0$, then the net force on the object must be (basically) zero as well. So a "massless rope" automatically exerts the same tension at both of its ends; but if you're given a problem where the mass of the rope is not negligible, this is no longer true.

Follow-up: it's important to note that Newton's Second Law is not terribly useful for objects whose mass is "truly" zero. It predicts that such objects must experience no net forces and no net torques, and doesn't tell us anything about their accelerations (since $m a = 0$ automatically, $a$ can be any number and still satisfy Newton's Second Law.) While you can predict the motion & rotation of very light objects using Newton's Laws, you can't say anything about objects whose mass is truly $m = 0$.

The only way to make meaningful predictions about the behavior of "massless" objects in Newtonian mechanics is to assume that their mass is non-zero, and then look at the behavior of the solution in the $m to 0$ limit. Depending on the system, this limit might be physically reasonable behavior, as it is in this case; or it might not be physically reasonable (e.g., a fixed force is applied to an object of mass $m to 0$.)