Alastor Moody took the position of DADA (Defense Against the Dark Arts) teacher in 1994. He was abducted by Barty Crouch Jr. at or before the start of the Hogwarts school year as part of the latter’s plan to bring Harry Potter to Voldemort . Is there any evidence that Moody ever actually taught DADA (or any other) classes at any time, either before or after his abduction?
No - Moody was captured before he could start teaching.
Barty Crouch Jr. captured Moody before Moody actually began teaching at Hogwarts. He and Wormtail captured Moody in his own yard, before the school year had started. Barty Crouch Jr. most likely did this because it’d be harder to get at Moody once he was teaching at Hogwarts.
“Forced him into a compartment of his own magical trunk. Took some of his hair and added it to the Potion. I drank it, I became Moody’s double. I took his leg and his eye. I was ready to face Arthur Weasley when he arrived to sort out the Muggles who had heard a disturbance. I made the dustbins move around the yard. I told Arthur Weasley I had heard intruders in my yard, who had set the dustbins off. Then I packed up Moody’s clothes and Dark detectors, put them in the trunk with Moody, and set off for Hogwarts.”
- Harry Potter and the Goblet of Fire, Chapter 35 (Veritaserum)
The “Moody” we see teaching at Hogwarts was always Barty Crouch Jr. and never the real Moody. After Barty Crouch Jr. was caught, the real Moody didn’t take over - Defense Against the Dark Arts was canceled, because the teacher was a fraud and the real Moody was in no shape to teach. Moody actually mentioned to Harry that it didn’t make sense calling him “professor” because he didn’t do much teaching.
“Professor Moody?’ he said uncertainly.
‘I don’t know so much about “Professor”,’ growled the voice, ‘never got round to much teaching, did I?”
- Harry Potter and the Order of the Phoenix, Chapter 3 (The Advance Guard)
He didn’t end up teaching Harry at any point during Harry’s fourth year, and his comment on the title “professor” could imply he hadn’t been one at any other point - if he had, he might be used to it.
Moody doesn’t seem to have taught at any other point either.
After Harry’s fourth year, he doesn’t come back to teach in another year. He dies in what would have been Harry’s seventh year, so he certainly couldn’t teach after that. He doesn’t seem to have been a teacher at Hogwarts any time before Harry’s fourth year either - he’s only ever mentioned as being an Auror before his retirement.
“Who is Mad-Eye?’ asked Harry.
‘He’s retired, used to work at the Ministry,’ said Charlie. ‘I met him once when Dad took me into work with him. He was an Auror – one of the best … a Dark-wizard-catcher,’ he added, seeing Harry’s blank look.”
- Harry Potter and the Goblet of Fire, Chapter 11 (Aboard the Hogwarts Express)
No one ever mentions him having taught at Hogwarts at any time prior to Harry’s fourth year. It seems like he was only ever an Auror until he retired, and he seems to have only became a teacher in Harry’s fourth year as a favor to Dumbledore, perhaps because Dumbledore wanted the students to learn from someone more practically experienced in Defense Against the Dark Arts.
As a Hogwarts Professor, no (as other answers have stated). But he did train Aurors if that counts.
"I don't know so much about 'Professor,' never got round to much teaching, did I?" - Moody to Harry in OotP
Tonks was described as Moody's protégée which implies he was actively involved in her training.
Clever, brave and funny, pink-haired Nymphadora Tonks was a protégée of Alastor ‘Mad-Eye’ Moody, the toughest and most grizzled Auror of them all.
He was captured before the start of term. Barty Crouch Jnr. spent the whole year impersonating him. This was only discovered after the Triwizard Tournament.
There was still some of the term left after this but it seems the real Moody didn't teach.
The only person apart from Ron and Hermione that Harry felt able to talk to was Hagrid. As there was no longer a Defence Against the Dark Arts teacher, they had those lessons free. - Harry Potter and the Goblet of Fire, Chapter 37, The Beginning