Had the same problem where Takemura calls you for the "Down the Street" job but then doesn't talk and stays there forever. My last save was right after a main job so I really can't progress further because Takemura's call will be there forever.
After numerous attempts to hack the bug, I found a solution that works, even if it seems odd.
The solution here is all about timing. Basically, what you need to do is have another person call you while being stuck with Takemura. The other call forces Takemura out of your screen and brings everything back to normal.
You can do this by carefully timing yourself walking into an area where Claire will call you for a side job.
So basically,
1. Takemura calls you, the bug happens because he stays there.
2. Walk into the area to initiate a call from Claire. Answer Claire's call.
3. Once you get Takemura out of the call, save your game right away and load it again.
There you go.
Though, this fix is a bit odd. Ever since I "fixed" it, whenever Claire calls me, Takemura is right next to her in the call like they're physically together but he doesn't talk. UPDATE: all your video calls will have Takemura on the screen taking over whoever you're supposed to talk to. Takemura is clingy.
P.S.
Not sure if this fix is good enough to continue playing the story. So just make sure you use a different save file for this fix.