When is it fine to drop to be when should precedes it? For example:
It should prove to be successful.
vs
It should prove successful.
When is it fine to drop to be when should precedes it? For example:
It should prove to be successful.
vs
It should prove successful.
Your question actually is not about the modal auxiliary verb should. Instead what interests you is the nature of the verb prove and what it allows.
Prove is a kind of copular verb, like seem or appear, which can link an adjective directly back to the subject as the subject's 'complement':
The project seemed/appeared/proved + successful.
These particular copular verbs also allow for an infinitival complement structure with the verb be:
The project seemed/appeared/proved + to be successful.
This latter structure with the infinitive is probably older and it does perhaps sound more formal or stilted than just the simple adjective.
We can tell the language is in the process of shedding these because some such copular verbs — other people on this page mentioned look and sound — are okay but a bit odd with the to be so we prefer them without:
- The cake looks stale.
- The trip sounds fun.
You may have also wondered about keeping or deleting to be in structures like the following, where the complement is linked to the object of the verb and not the subject:
We elected him (to be) treasurer, but we found him (to be) utterly incompetent!
For questions like these, look to the nature of the main verb for answers and patterns of use. The auxiliary verbs that fall in front of the main verb won't govern those patterns.