(At François's request, my comment in now an answer.)

Yes, it is still an active research area. It however is spread out throughout a number of camps (traditions): The Weihrauch camp, the reverse math camp, the computability theory camp, the randomness camp, the proof theory camp, and a few different constructive math camps. (Also, see many of the quantitative results in classical analysis.) Most researchers span two or more camps, and I don't mean to imply there is a feud or anything. However, there isn't necessarily an organized central list of open problems or a central agenda. You may want to check out cca-net.de as a starting point.

(At François's request, my comment in now an answer.)

Yes, it is still an active research area. It however is spread out throughout a number of camps (traditions): The Weihrauch camp, the reverse math camp, the computability theory camp, the randomness camp, the proof theory camp, and a few different constructive math camps. (Also, see many of the quantitative results in classical analysis.) Most researchers span two or more camps, and I don't mean to imply there is a feud or anything. However, there isn't necessarily an organized central list of open problems or a central agenda. You may want to check out cca-net.de as a starting point.

Edit: You may also want to check out this survey by Brattka and Avigad. It does a good job of explaining many of the different traditions.