I'm buying a commercial printer that has a 16 amp load, so I'll be installing a 20 amp circuit breaker. How do I know if I will need to upgrade the wiring for this change? My house was built in 1950, so the wire shouldn't the wire all be heavy enough? Or not? I'd like to think I only need to replace the breaker, but...

I'd like to think I only need to replace the breaker, but...... But, I don't think you're going to be that lucky. :(

It will work on a 20A circuit, but that one thing will max out the circuit when it is being used.
I HIGHLY doubt you will have the circuit you need where you need it.
A dedicated 20A circuit would be the best thing for this printer, and you.