From the comics: Dr. Strange has been shown paying the bill for use of magic on multiple occasions.
First: in the comics, the Eye of Agamotto is not the Time Gem, and I'm not sure the Eye has been shown manipulating time (though it may well have been used to view past events). So, no, the comics do not show him "paying the bill" for manipulating time, because that's something specific to the films.
That said, the idea that the good Doctor might have to pay for his use of magic is not a new one.
When Stan Lee first created the character, he simply made up cool-sounding names for different magical spells Strange would cast: the Rings of Raggador, the Crimson Bands of Cytorrak, etc. After a year or so, he took one of those entities Strange had named, Dormammu, and turned it into a character - and one of Strange's deadliest opponents. (Side note: Cytorrak quickly got an additional mention as well, as the source of the gem that gave Cain Marko the power of the Juggernaut). Note that after fighting Dormammu, Strange never tries to use him in an incantation again.
In the 1980s, there was a storyline (when Strange didn't have his own book - he starred in one-half of Strange Tales, with Cloak and Dagger in the other half) where Strange was forced, for various reasons, to use some darker magic. He lost use of the Eye of Agamotto, and found himself driven more and more to use dark magic. He finally turned things around, and was able to set aside the dark magic and go back to his old self.
In the 1990's, the various entities Strange had called on (the Vishanti, Valtorr, Raggador, etc.) called for him to pay them back for their help over the years by participating in a big magic war. Initially, he tried to find other magical paths (he changed his magickal style at least once a year for a while). Finally, he agreed to serve in the war. Due to magic, although we fought in the war for decades, he came back able to use his magic as he had in the past, and looking the same as always.
Finally, in Jason Aaron's recent run on Doctor Strange, we see that that Strange eats is mostly bizarre, disgusting stuff that they wouldn't have used in the Fear Factor TV show, as at least partial payment for using magic. Plus with most magic removed from his reality, he's searching for relics and artifacts to regain access to magic, rather than tossing off "bolts of bedevilment" and such. That's why he's running around with an axe now; he doesn't have enough magic to do what he used to.