Which Yongnuo flash is compatible with an Olympus Pen F? Do I need to purchase Yongnuo triggers to use the flash? I currently have other brand triggers and would like to use those.
Which Yongnuo flash is compatible with an Olympus Pen F?
None of the Yongnuo flashes will fry your digital Olympus mirrorless camera. But none of them are fully-compatible with it, either. Yongnuo at this time only makes TTL/HSS gear for Canon and Nikon (with one TTL/non-HSS flash for Pentax). Their manual-only gear can be used with Olympus, but there is no TTL, HSS, 2nd curtain, etc. capability. Just firing in sync, and (possibly) remote power/zoom control, if you use Yongnuo manual-only flashes with built-in radio triggers (YN-560 III, -560IV, -660, -720), and the corresponding Yongnuo transmitter (YN-560-TX).
Do I need to purchase Yongnuo triggers to use the flash?
If you want remote power/zoom control, yes. You need to use the YN-560-TX and the Yongnuo single-pin manual flashes with a built-in radio trigger.
I currently have other brand triggers and would like to use those.
You can use them with Yongnuo flashes, but then all you get is firing the flash in sync.
I would highly recommend (at the time of this writing) that you consider, instead, going with Godox gear (both flashes and triggers). With a TT600/X1T-O combination (which costs roughly the same as a YN-560IV+YN-560-TX combo), you not only get remote power control, but also HSS from an Olympus camera. With a Godox TT350-O or TT685-O, you'd also have remote TTL and zoom control. Godox supports micro four-thirds as a system, and the expansion options are much better than Yongnuo's.