Assalaamu alaykum. I pray in a masjid on the third floor of a multipurpose building. There is a lift in which it is written that non-residents are not allowed to use it.
1) Now, as a non-resident, will I be sinful if I use the lift?
2) Sometimes, if I do not use that lift and use the stairs instead, I may miss the first Rak'ah (unit of prayer) of the congregational prayer, am I allowed to use the lift in such case?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger. 

Since the owners of this lift prevent others from going up in it, then you have no right to use it without their permission, because the money of a Muslim is not lawful except with his own proper will.

Therefore, you have to take the stairs, even if you miss something from the prayer.

The virtue of catching up with the first Rak'ah of the prayer should not be attained by doing what is forbidden. Also, you may go somewhat early in order to attain the virtue of the beginning of the prayer.

Allah knows best.