I have a traditional IRA and Roth IRA in Fidelity for backdoor Roth purposes. I plan to rollover my old Roth IRA from Betterment to Fidelity. Will this cause tax issues?

Although it's possible to move a Roth IRA by distribution and rollover, if the only purpose is to move the funds, it's much preferable to do the move by nonreportable trustee-to-trustee transfer since a trustee-to-trustee transfer avoids any involvement with the one-rollover-per-12-months limitation.  A trustee-to-trustee transfer of an IRA is neither a distribution nor a rollover.

It's usually best to initiate the transfer at the receiving custodian, Fidelity in this case, so the the receiving custodian can identify to the original custodian the Roth IRA account to which the funds are to be transferred so that the original custodian can make the funds payable to the correct account.