I live in an apartment building and my lease is up on may 31, I needed to give notification in writing if I was not going to resign my lease, I did not do this nor did I resign my lease. Can I still stay living in my apartment

Most leases merely renew, you don't really sign a new lease, the original one merely renews as valid for the next coming year, If this is the case, by not giving notice, you are in a new lease now anyway.

If you are suppose to sign a new lease and do not, then any default provisions of the old lease could be put in place.

Also normally if a lease ends, the rent goes to a "month to month" rental and those provisions in the lease shows those terms