Can a dentist pull a tooth if it's infected?

Unlikely: Pulling teeth in the presence of a gum infection isnt recommended, and pre-treatment of the infection before tooth extraction is required. This is because removing the tooth when the gum is infected can make the infection spread.

Technically yes: But it is much better to wait for the infection to be brought under control.

Yes: See your dentist and (s)he will treat or pull the infected tooth

Yes: Sometimes tooth should be removed immediately. Sometimes antibiotics required first to control the infection. Sometimes root canal is a better option. See your Dentist for an evaluation and customized advice.