The original program is open sourced under GPL v3, but asks for a fee to use it. I did not pay for it, but I'd like to share the no-fee version I compiled with my college friends (obviously under GPL v3). Can I do that?
The GPL allows any use, including commercial use. And the author or distributor is not required to provide sources to anyone, but only to those which he distributed to the programme in binary form.
However anyone who obtained the binary of a GPL-licensed programme legally, is entitled to receive the source code. And the GPL license allows to make modifications as long as they are clearly marked and attributed and the modified version does give credits whom credits are due (thus copyright notices must be retained, but can be ammended).
Thus if you obtained the source code under a GPLv3 license, you are totally allowed to modify it, use it in compiled and/or source form as you see fit. You can also distribute it as long as you abide by and fulfill your obligations as stated in the license itself, namely to also make available the source to your modified version at least upon request to the recipients of your modified binary version.
That would be difficult, if you just hacked the executable instead of modifying the source and compiling that - and then you'd probably be in a grey area.
Mind though, just because a previous version of a programme was licensed and distributed under GPL, does not mean that the currently-distributed version by the original author is distributed under GPL. Whether or not a license change is allowed depends on who holds the actual copyright.
A famous example where a license key restriction was introduced in GPL-licensed code is rhodecode which hence was forked under the name kallithea which removed the license restriction introduced at that point and kept the programme free and open-source. Since then, rhodecode reverted its decision at least partially.
tl;dr: Yes, you can.
"The original program is open sourced under GPL v3, but asks for a fee to use it."
This doesn't make any sense. If the program was distributed to you (in either binary or source form) under the GPL, then you have a license to use it without paying any kind of fee. Once the software has been licensed to you under the GPL, this permission cannot be revoked unless you violate the terms of the GPL.
From Section 2 of the GPLv3:
All rights granted under this License are granted for the term of copyright on the Program, and are irrevocable provided the stated conditions are met. This License explicitly affirms your unlimited permission to run the unmodified Program.
You are additionally allowed to change it and use said changed version without any further restrictions:
You may make, run and propagate covered works that you do not convey, without conditions so long as your license otherwise remains in force.
You also have a license to distribute said changed version of it to whoever you want without paying any sort of fee, so long as you continue to license said distribution under the GPL (same version or newer) and preserve the copyright notice. If you distribute a compiled version of it, you're required to also make the source code available to whoever you distributed it to, also under the terms of the GPL. These rights are covered in Sections 5 and 6 of the GPLv3 (or Section 4 if you're distributing a verbatim [unmodified] copy.)
It sounds like whoever distributed this to you under the GPL didn't understand the GPL. Or hoped you didn't.
So, in short, yes, you can modify it to remove the part asking for payment and you can additionally distribute the modified version of the work - in either source or binary form - to anyone you want, so long as the distribution remains under the terms of the GPLv3 (or newer.)