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?

[Opus 17]

Rolling over a Roth IRA from one trustee to another has no tax consequences if done correctly.  You should have the transaction done directly by the two trustees, instead of getting a check yourself that you need to forward to the new trustee.  

[TomD8]

Technically, moving an IRA account from one custodian to another is an IRA transfer, not a rollover.  This web article details the IRS rules regarding IRA transfers:

https://www.investopedia.com/terms/i/ira-transfer.asp

[dmertz]

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.