Just happened to me as well; switching to an alt and back cleared the error, but still a nuisance
Year 2021 and still happening. Logging out to an alt and then back does the trick.
Happened in Westfall. Relogging worked. Indie game
this is still happening
confirmed still occurring (Terrace of the Collectors, Bastian for me) - confirmed logging onto alt and back again resolved this.
Still happening as of April 13th 2021. I have a feeling this will never get fixed 
I just got this bug.
What Happened: Logged on > hit a ‘Raid Target Dummy’ > Did daily anima table > Flight Master Gave me error.
How I fixed: Guessed that maybe there was a bug on 'Raid Target Dummy" that puts you in some form of Combat, so I ran and fought it till it killed me. Respawned, issue fixed.
Good god this thread has been necroed a lot of times…
In any case, this tends to happen if you get in combat with a mob, then the mob gets into combat with something else. Even if you leave combat, the mob that’s in combat with something else will still prevent you from using flight paths.
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happening right now after downing valinor. Can eat, can mount up, tried reload ui and logging. Nope.
edit: thankfully hearthing helped, so maybe it’s specific to that NPC, maybe a flag/checkbox, who knows. Not gonna go all the way there to test that theory.
Happening to me right now leaving Terrace of Collectors in Bastion, can’t use my teleport pad back to Bastion’s covenant sanctum. Can mount, can’t use flight master, can use hearthstone and that fixed it. May 10th , 2021
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This is the place it keeps happening to me in Shadowlands and it drives me nuts. I’m not technically in combat because I can mount, eat, etc, but Thrill Seeker (Venthyr soul bind ability) that is only activate when you’re in combat is proccing like I am in combat. Leads me to believe the game still registers you in combat even though you aren’t.
EDIT: While I was sitting here writing this post I somehow dropped “combat” and was able to take the FP. Couldn’t have been longer than ~5 mins.
Happened to me just now and I was in a mob, got away and now here I am , no taxi service. I could wait but it’s been more than 5 min so I opted to use the suggestion above to sign onto another toon and take a flight path, then signed back on to my mage with no issue.
This bug is still going on and by the looks of this thread will continue to exist for quite some time.
I always/only run into this issue when killing the world boss valinor. I mount up and go back to the flight point and that’s when I’ll get the taxi service can not be used cause im busy blah blah Can still mount up and hearth.
The only solution to fix this is to relogin to the game.
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Yeah, I just got this bug after killing Valinor, logging into another character and back on fixes it.
Happening June 24 2021.
Things I did just prior … smelt ore. Look at cooking recipes. Mount up. Run to FP. Dismount. Try to fly.
thanks serai and it still happening 2021 on dun morogh quest stricking back
I just had this happen to me in Bridgeport. After I read posts on here for a bit, I could us the flightmaster.
24/01/2022 still not fix. Just happened to me. Dc, reload interface, disable addon dont work. Log off, log on an other character and relog worked.
April 25, 2022,
The Horde Flight Master refuses me passage, mentioning to come back when I’m not so busy. I tried showing the guy that I was, in fact, NOT busy by sitting and drinking some delicious melon juice, but he still insisted I was busy.
(using /reload ui worked for me, though)
Still happening in sep 2022.
Reload ui didn’t work, logging out did.