I was once told that there is a word for when a movie's(or any work of
art) title is said during the film. What is that word?

Hello.

The term is "title line."

"...A title line is one of those times where the character says the
title of the movie in the movie--preferably at the emotional climax. "
source: Ewerx forum
http://ewerx.ca/forums/viewtopic.php?p=1063

Usage of this term may be seen in various movie reviews:

"Review: He Got Game...
...Michael Jordan, who utters the title line in a testimonial about
the young Jesus. "
source: talkcinema.com
http://www.talkcinema.com/archive/he_got.html

"Pucker Up and Bark Like a Dog (1990): 
...Wendy O. Williams as a sexaholic biker (she says the title line)"
sourcee: joebobbriggs.com
http://www.joebobbriggs.com/drivein/1991/prisonstorieswomenontheinside.htm

-----

search strategy:
"character says the title"
"says the title line"
"utters the title line"

I hope this helps.

Thank you, that's been bothering me for a long time.

It's actually called "titular line"

Yes, "titular line" is also used, but Google searches indicate that
the terms "title line" and "titular line" are used interchangeably.

"the title line" movie
://www.google.com/search?q=%22the+title+line%22+movie&hl=en&lr=&ie=UTF-8&start=10&sa=N

"the titular line" movie
://www.google.com/search?hl=en&lr=&ie=ISO-8859-1&q=%22the+titular+line%22+movie&btnG=Search