Going from 2 chainrings to 3 on my Crankset, do I need a new front derailleur?

It isn't only the travel of the front derailleur. If the cage isn't long enough then the chain may not be able to reach the smallest chainring without riding on the cage.

On top of that, the rear derailleur may not be able to take up the extra slack in the chain which means the chain will sag and fall off the chainring or rear cog very easily. You will likely need a long (or mid) cage rear derailleur.

You will likely need a longer chain. If you adjust everything to compensate for a smaller chainring the chain may be too short to get over the big chainring. If you adjust for the big chainring the slack may be too great for the derailleurs to handle.

If you are changing only the crank and not the bottom bracket you might also discover the inner chainring might contact the frame. In order to avoid contact there should be at least 4mm clearance between the frame and closest chainring.

Finally, unless you have a non-indexed shifter then you'll also need a shifter that is made for a triple.

Your current derailleur may not work properly with a triple, so you may have to change it. You will also probably need a new shifter. Had you included information about your current components, it might have been possible to tell for sure.

So, the answer is "Mybe." The old one probably does not have a detent in the center position, which is a big help in switching to the middle range.

depends

probably not

though shifts that you should not do anyway [google cross chaining bike] might not work well, or make a noise or something undesirable but not fatal

wle