If L=P then P is not equal to PSPACE. This follows from PSPACE properly containing L.
I am wondering if L=P implies any stronger separation between complexity classes? Does it imply P is properly contained in NP?
As Tsuyoshi points out, it is consistent with current state of our knowledge that L=NP in which case L=P=NP.
The question can be state more rigorously as improving the result in the first line of this post:
Is there an interesting complexity class C which we don't know if it is separated from L (by the space hierarchy theorem, etc.), however we know that if L=P then L will be strictly contained in C?