I remembered one scene in TOS, in "Balance of Terror", where the Enterprise travels backwards at warp speed, and I hoped to find others.
Using the search function in the site for searching Star Trek transcripts from all shows and movies from TOS up to Star Trek: Enterprise, I found numerous references to "astern", but mostly to describe the position of other vessels relative to the one the protagonists were on.
http://scriptsearch.dxdy.name/1
I found several references to the Enterprise NCC-1701 and the Enterprise NCC-1701A travelling astern, but there is only one case when warp drive is mentioned, the scene I remembered:
In TOS "Balance of terror" the Romulan ship turns off its cloaking device to fire at the Enterprise:
(The Romulan ship has become visible)
KIRK: Full astern! Emergency warp speed!
(A red plasma blast is heading towards them)
KIRK: Do we have emergency warp?
SULU: Full power, sir. It's still overtaking us. If we can get one phaser working, sir, one shot might detonate it.
KIRK: Navigation?
STILES: Estimate it'll overtake us in two minutes, sir.
KIRK: Phasers, Mister Spock?
SPOCK: Impossible, Captain.
KIRK: Feed this to the space recorder and jettison immediately.
RAND: Captain, should I continue log entry?
KIRK: Yeoman. Affirmative. Continue log entries.
RAND: Yes, sir.
SULU: Ten seconds to impact. Captain, It's dissipating, sir.
STILES: It must have a range limit.
(Rand stands as close as humanly possible to Kirk)
SULU: Five, four, three, two, one. Impact!
(Everyone gets thrown about a bit)
http://www.chakoteya.net/StarTrek/9.htm2
So instead of trying to turn the Enterprise away from the Romulan energy bolt Kirk orders full speed astern with emergency warp speed, and Sulu later says they are using full power emergency warp - presumably at least warp factor eight.
So "Balance of Terror" establishes that in the era of TOS at least some Starfleet starships, including the USS Enterprise NCC-1701, can travel backwards at warp speeds up to full emergency warp speed.
It is possible that in the era of TNG Starfleet starships are no longer built with that function, but since they seem to be more advanced than TOS era starships in most respects it seems unlikely that they are no longer able to use that specific function.
And as far as I know from the search the only fully canonical example of a sStarfleet Starship travelling backwards at warp speed is in TOS "Balance of Terror". It is probable that Starfleet starships retained the ability to travel backwards at warp speeds in TNG, DS9, and VOY, but never had reason to use it.
But I cannot prove from canonical evidence that TNG era starships retained the ability to travel backwards at warp speeds since there is no canonicla evidence that TNG era starships ding it.