Do you need a mixing board to record a podcast?

Simply put: no, you do not need a mixer to start podcasting. But there are reasons why you should look at getting one:

Recording with multiple microphones for guests
Recording a Skype or phone call
Greater control of sound
Better sound quality

If your podcast has more than one host, or if you're taking regular calls then a mixing board is recommended. Otherwise, try a few episodes without one and look to get one in the future.