electron donation affecting carbonyl bond strength
Let's take the above molecules, acetyl chloride and ethyl acetate, as examples of acyl compounds with electron-withdrawing and -donating groups, respectively.

We know that Cl has a higher electronegativity than O as a halogen, so the electron density from the carbonyl carbon should be pulled toward the chlorine in acetyl chloride and pushed toward the carbonyl oxygen in the case of the ester. This might not be immediately obvious, as the two oxygens should have the same electronegativity, so I like to relate this latter point to electric force laws: 
$$ F=\frac{kq^2}{r^2} $$
Since C=O has a shorter bond distance than C-O, the C=O will exert a greater force on surrounding electrons than the C-O (this assumes electrons to be point charges, so it is only useful as a trend).

Therefore, we draw the following picture:



Meaning that there is more electron density on the carbonyl carbon for ethyl acetate than for acetyl chloride. 

Now, that's somewhat of a problem for the carbonyl oxygen of ethyl acetate! It already has a lot of negative charge as an oxygen, so it will experience repulsive forces due to the increased electron density on its adjacent carbon. Remember that oxygen does not like having more than a charge of -2 formal charge, which is already satisfied by the double bond it has to the carbonyl carbon; a similar relation can be made for actual charge density. 

These repulsive forces, well, repel the oxygen away from the carbon a bit, increasing the C=O bond length. And we also know from physics that increasing the bond length decreases the frequency of vibration. This should make intuitive sense: if the bond distance increases, then the atoms have more distance to travel to complete one cycle of vibration, thus decreasing the number of cycles that can be completed per unit time.

Now onto our example electron-withdrawing group:

Acyl chlorides pull electron density away from the carbonyl carbon, as mentioned earlier. Following the same logic, we can conclude that the C=O bond distance decreases(... to a point*). It follows that the C=O vibrational frequency should increase.

Well, that was a lot of words. But let's approach this from another, much simpler, way. In an acyl chloride, the chlorine acts as an electron-withdrawing group, destabilizing the already electrophilic carbonyl carbon. While this decreases the stability of the molecule, that does not mean that all bond strengths are diminished. In acyl chlorides, the C-Cl bond is the weak, labile bond, not the C=O bond. In fact, the C=O bond strength increases for the reasons outlined above. In our example ethyl acetate, the C-O donates electron density to the electrophilic carbon, thus stabilizing the compound. This does not mean that all bonds are strengthened. If anything, it might indicate a greater C-O bond strength than would otherwise be expected. The C=O bond however,  is a different bond that is being affected inductively, so it must be considered differently.

Personally, I am a fan of the first explanation, as that provides an actual understanding of what's going on, whereas the second explanation leads to a more memorization-focused approach to chemistry. I hope this helps!



*Extra information--Let's consult the following graph (ignore the specific numbers on it):



What this shows is the distance between two atoms and the potential energy of their "bond" (this is one of those areas where it should be noted that bonds are really just constructs to help us make sense of molecules... but this is not important for now). To the right of the minimum, attractive forces (nucleus --> electrons) dominate. To the left, repulsive forces (nucleus &lt;--> nucleus) dominate. As two positive charges get further away from each other, their energy goes to zero (this can similarly be rationalized by simple physics II-type equations). As their distance approaches 0 (where a distance of 0 indicates superposition, or direct overlap), the energy between the two atoms goes to infinity. This is all just to say that there's a limit on how close the atoms can get to each other. You can see that the slope to the left of the minimum is much steeper and increases at a much faster rate than the slope the the right. It follows that there should be less difference in the C=O bond distance as compared with some fair reference molecule such as acetaldehyde than between the ester and that reference. Part of why I'm bothering with this at all is because of how useful this graph is in just about everything I have studied in some shape or form, so it is very useful to familiarize yourself with it.