Can I use a 3.7V battery for a 3.3V board?

You can use an ultra-low-dropout linear 3.3V regulator. Fully charged LiPo are 4.2V, beyond the abs max for many 3.3V chips, the consequence of not reducing the voltage is almost certain failure.

LiFePO4 cells are 3.2V nominal and charge to a maximum of about 3.6--3.7V so are usually fine with 3.3V chips. They are safer than LiPo too so a pretty good choice all round, apart from needing a particular charger.