I have a 15 amp breaker that occasionally flips off, especially when we have an iron and heat lamp on. Can I just replace that 15 amp with a 20 amp?

What is the wire size?

Unless you know what gauge the wire is, no. I will assume the wire is 14AWG which is rated at 15 AMPS. You have over that with and iron and a heat lamp. IF you can verify the wire is a 12AWG, you may replace the 15 with a 20. Try moving either one to another circuit to reduce the load.