Over the last 30 or so years that I have been working on electronics I have used just about every method there is for removing through hole ICs. YES, you can easily burn an FR4 PCB. I see it all the time. Burning a PCB can/will carbonize the PCB, which is conductive. Any burned fiberglass must then be scraped out and repaired. On multi-layer boards this damage can be terminal as carbonized PCB in middle layers (often power and ground planes) is almost impossible to repair. I have seen a lot of PCBs destroyed when someone got careless with heat trying to "repair" boards.
The best method I have found for through hole components is using regular old SnPb solder and an Edsyn Soldapullt. The key is to first add a bit of solder to each pin first, then heat (no more than 2 seconds unless it is on a ground plane), then immediately suck solder up. If you don't get it clean the first (or second or third) time, add more solder and try again until the barrels are clean. Keeping the pins straight and wiggling them slightly with the tip of the iron helps as well. If the pads/pins are dirty/previously burned/corroded you have to clean them up as good as possible before you start. Once all the through holes are cleaned out, I twist the IC to see if it is loose, repeat the desoldering procedure on any pins that are stuck (or just heat with the iron and pull if it is just one or two - usually on ground planes) and pull out the IC. I can now remove just about any through hole component without damaging the component or the board.
You may find that with badly burned ICs where the solder is burned or ICs that definitely do not need to be saved, it is easy to cut each pin off the body of the IC, add a bit of solder, then pluck each pin out one by one with the tip of the iron and desolder with the SoldaPullt (best solder sucker ever!). This method takes a bit longer than just desoldering, but it is safer on delicate PCBs.
For what it is worth, I almost never use solder wick when removing components. About the only time I use wick is to clean surface mount pads or remove solder shorts on surface mount parts.
For my soldering station, I use a MetCal MX500. I have used lots of different irons, but once I found MetCal I was done. Due to the design, they heat up virtually instantaneously. I have about 20 different tips for the normal handpiece, a Talon (tweezer like tool) that is useful for some surface mount parts and the air powered MetCal desoldering tool. Although it works pretty well, it still isn't as good as a SoldaPullt and is much harder to clean.
Just my $.02, YMMV.