Summary
Looney had some nice years years with the Warriors from 2017-2019. In the '17-'18 season, Looney averaged just under 13 minutes per game, but that ramped up to over 17 minutes per game in the playoffs, and he outplayed some of their other options at center (not counting Draymond Green). However, as those centers were Jordan Bell and Damian Jones, the options were pretty limited. In '18-'19, Looney averaged just under 18 minutes per game and played 80 games, and his minutes went to over 20 per game in the playoffs.

However, after missing most of the '19-'20 season due to injuries, and not looking quite like his old self physically in '20-'21, Looney had a bounce-back year in '21-'22, and played great for the Warriors in their 2022 playoff run.

Looney won't wow you on either end, but he plays within himself and has earned the trust of Steve Kerr, starting the vast majority of games for the Warriors in '21-'22. He doesn't have much scoring game, but he doesn't take bad shots and understands how to play within the Warriors' system. He also has great hands, both when it comes to catching passes in traffic and gobbling up offensive rebounds. His 13.9% offensive rebounding rate in the playoffs was awesome, as he consistently punished opponents on the glass.

Looney isn't super quick laterally, but teams made the mistake of thinking they could punish the Warriors by trying to attack him on the perimeter. In the playoffs, he had some great moments defending Luka Doncic and holding his own. Overall Looney just tries hard on defense and generally knows where to be, as he has been a key cog in the Warriors' elite defense.

Looney is an interesting free agent in that he probably since I'm not sure how valuable he would be to any team besides the Warriors, so it may be hard for him to create external leverage when negotiated with the Warriors. Looney is still only 26, so teams might have some interest in him as a low-end starting center or high-end backup.
 
Cap Considerations
Looney will be an unrestricted free agent with a $10.2 million cap hold and full Bird Rights. As his cap hold exceeds the salary he will get in free agency, his cap hold is not really relevant. Since the Warriors will have Looney's Bird rights, they will have no restrictions on re-signing him.

However, given the Warriors' high luxury tax bill, re-signing Looney could cost them exponentially in luxury tax penalties. Therefore, other teams will have a chance at prying him away from Golden State without a huge offer.

Other teams will have multiple ways to sign Looney as even teams without cap space could possibly use either the Non-Taxpayer MLE or Taxpayer MLE to sign him. There's even a chance he could be had for the Room MLE or Bi-Annual Exception, but I think that's unlikely given how well he played in the playoffs.