Desoldering DIPs is not that easy. Even if you have removed all the solder from each hole, there is usually a tiny bit of solder remaining between the pin and the edge of the hole, forming a union between the pin and the hole's copper plating.
I found, many years ago and to my dismay, that it is very easy to pull the copper plating out of the holes along with the pin on the Mac 128 boards.
If you are certain that the chip needs replacing, then you can clip the pins off of the body of the chip, and then carefully desolder the pins one at a time. Take care not to pull on any pin with any force until it is desoldered and will come loose easily.
Alternatively, turn the board over, cover the areas surrounding the chip in question with modeling clay (cheap and readily availble) and then use an adjustable heat gun set to about 700 - 800F to melt all the solder on all the pins simulataneously. At that temp setting it takes about 2 - 3 minutes of heating for the chip to be ready to drop out with my heat gun. Others will vary of course.
Part of the heating is heating the copper layers inside the circuit board. Until those are warm, nothing is going to melt. If you had a second chip to desolder, the second one would go much faster.
An implement such as a dental pick is handy here, because as you're heating the solder, you can poke the solder in each hole to see if it has melted yet. This also lets you figure out if you've melted all the holes on one side and none on the other side, e.g.
Once the solder is melted in all the holes, you should be able to push the chip out by the pins, using the dental pick. Do not pry from the other side with a screwdriver--that's a recipe for prying on the chip before the solder is melted, and again, leads to empty pin holes.
Of course, if you have a hot air soldering station, then that's much better, but my specialty is doing difficult soldering with bits you can buy for under $40 at the hardware store and Radio Shack.