Do I need to apply for a tourist visa before my third entry into Thailand?

There isn’t a limit as such on visa exempt entries. That said you won’t be able to get unlimited entries. At some point the immigration officer will either warn you to get a correct visa or flatly deny you entry.

No one can say after how many entries that could be as it’s up to the individual officer you give your passport to.

If you’ve maxed out two entries and extensions then there is a fair chance you could have an issue.

You’ve posted in the group about the DTV Visa so look and see if you qualify for that. From many reports it’s a relatively easy option and would give you 5 years of entry, albeit limited to 180 days at a time.