I'm sending my iPhone 5 back to be worked on, do I need to remove the SIM card before I send it back?

no unless any instructions state otherwise.

Please do not send any accessories with your iPhone because they will not be returned to you. If your iPhone has a SIM card, make sure to remove it prior to service.

Extract from the Apple guidance on mail in service 😉

AND as you will sent back an exchange unit if possible make sure you back up remove any data etc as you will not see your original device again

Preparing your iPhone, iPad, or iPod touch for repair or return

If possible, please perform these steps in the following order to prepare your iPhone, iPad, or iPod touch for repair or return:

1. Sync your device with iTunes.
2. Back up your device.
3. Disable iMessage: If iMessage is active, please turn off iMessage. Tap Settings > Messages > iMessage.
4. Erase all data from your device by tapping Settings > General > Reset > Erase All Content and Settings.
5. Remove any case or film, if present.
6. If your iPhone or iPad has a SIM card, remove the SIM card.

Note: If your device has cellular service, be sure to contact your carrier to suspend service. You can also discuss what options may be available for using another device while yours is being repaired.