Because of the issue we had with our old heads we have to find new heads for a 89 firebird 305 TPI. The heads that were on it was casting #081, and the only heads that I can at a shop around my area is 187. Will casting #187 heads run on a 305 V8 TPI?

Yes they will work. They are both center bolt valve cover heads with 58cc chambers. The swirl port 187 heads were found on LO3 (305 TBI) cars. They flow less on the intake than 081's but flow more on the exhaust. They are fast burn heads so you will not need to run any initial timing advance (TPI cars had around 6 TBI had 0). The car may run funny on the stock tune since these heads will change the overall timing needs of the combo.