How can I remove the music from a Hallmark recordable card with music?

Possible, yep. Practical? hmmm.. depends on whether you have a microchip disassembly lab and a laser scalpel handy.

The last time I took one of those apart it boils down to the battery, the ‘speaker’ and one chip.
The chip is hermetically sealed in a blop of plastic so first tricky step would be to get that off without damaging anything.
Next step would be to identify the chip and read what you can out of the microcontroller’s program. You might be able to discover a branch to the ‘music’ the following playback of your recorded sound.

Next step would be to either disable that branch or fill the ‘music’ data full of zeroes or some value that represents no sound. Unfortunately that part of the chip is probably hard-coded in un-reprogrammable memory. If you can get the case off the chip and determine where the data are you might or might not be able to laser-zap the values you want into the fuses and circuits in the chip – depends a lot on its architecture.

It is possible – but I doubt it because it would be use a more expensive chip – that the program and ‘music’ section are reprogrammable in which case you just pull out your microcontroller programmer rig and blow out the branch to the ‘music’ with some no-ops.

Then put the case back on and replace the plastic goop – easy peeezy !

Sounds like a fun challenge but I’m sure I lack both the tools and skill for such an accomplishment.