Can the verb 'be' be modified?

To be certainly can, because it has its existential meaning, as well as its copulative meaning.

To painfully be, not to painfully be.

I find it interesting though that while this suggests more motive for the dilemma than the original it weakens it not just in ruining the scansion (it's not like I thought I could improve on Shakespeare) but because the existential meaning is so much rarer than the copulative that the misinterpretation "to be or not to be what?" is easier to make here. (We could explore the effect of the decision as to whether or not to split the infinitive on various phrases that modify the existential to be).

Still, used existentially, be can clearly be modified, since it is a normal enough verb in function.

For a stab at modifying to be in it's copulative sense I would try:

I still am.

We gladly are.

In leaving out the predicate (having it deduced from a context that I don't give), I think it shows that what is modified is the link between the subject (I) and that missing predicate; that is to say, what is being modified is precisely what the copula represents.

This doesn't give a complete answer:

1. We could read gladly above as modifying the verb are, giving us an answer of "yes".
2. We could read gladly as modifying the verb phrase "are [object or adjective deduced from context]", giving us an answer of "no".
3. We could read it as the second reading above, but conclude that that entire phrase is contained in the single verb, giving us an answer of "yes, sort of".

In the end, I'm left to conclude that whether we consider gladly as modifying are above comes down to why we are asking the question, we have different models for how language generally, and English specifically, works to solve different questions about it. It suits my needs (to have my hastily written pieces reasonably coherent and my more carefully drafted pieces coherent and convincing or evocative) to just answer "yes", but I won't claim to know how well this would fit with various models of grammar used by linguists.

Still, you can definitely modify the existential sense.