Can I remove a GSM module from an old cell phone?
Does connection cables exist for your phone? Or, if it is a Nokia, are there pins behind the battery?
If yes, your old phone is probably able to receive AT or FBUS command, allowing you to easily use the internal modem, for example to send/receive sms.
Look at this page for information about the FBUS protocol (for Nokias). This other page deals with AT commands to send an read SMS (This should work with most phones).
Search about this (for example on chiphacker), many people have done such things.
There also exists standalone GSM modules (example at SparkFun).
Unfortunately, no.
In cell phones, the RF components are very tightly integrated with everything else.
Basically, for size and cost reasons, they design a custom circuit board with the GSM interface devices and the actual phone hardware all assembled together.
It may be possible to tap the interface between whatever local processor the phone uses, and the RF stage, but it would very likely not provide a simple interface similar to the module used on the linked page. In any event, it would be a significant challenge to even get access to the interface wiring, as most phones use PCBs with many layers (4-10), and all the parts are generally BGA.
In addition to this, you would also likely have to do a lot of processing to get the RF section of the phone to properly communicate with the cell towers, since it's likely that the phones main processor handles some portion of the wireless interface protocol. This would require a fairly powerful MCU and non-trivial programming effort.
Basically, there is a reason they sell phone modules like that.
@Klaus's idea about using the phone as a whole is a much better idea. No PCB hacking there.
If some nokia's have a built-in AT command interface, and you have one of those phones, you should be good to go. Effectively, the whole phone is one of those GSM modules.