I assume you're using .012 or .013 gauge strings if you're tuning to drop B. There is no significant danger of the strings breaking by tuning up to E standard, BUT as mentioned in comments this will put extra tension on the neck and as you've already noticed on the tremolo bridge as well. Basically, if you want to tune up to E you'll need to do one of two things:
Do a new setup on the guitar with the current strings. This involves tightening the springs to bring the bridge back down to rest position, adjusting the truss rod, and resetting the intonation.
Change strings to a lighter gauge (.010 - .008) and tune to E standard. This will also require a new setup, but probably not as drastic as would be required in scenario 1.
If you are unsure about how to do these things, either get your guitar teacher to teach you or have it done by a professional. If you're planning on switching tunings frequently, you'll probably do:
- What most of us guitarists do and buy another axe. I have a guitar with .013s on it that I play in C standard/drop Bb, a couple that are in E standard/drop D (one has .008s, the other has .009s), one in Eb standard, and a 7-string in B standard/drop A.
At some point I'll need buy a second 7-string to tune down to F# standard/drop E (unless I buy an 8-string instead).