I'm currently reading book 2 of The Kingkiller Chronicle, and the series' main character Kvothe spends a lot of time worrying about the cost of tuition at the University. And while there are a lot of similarities between the University and A Song of Ice and Fire's Citadel, I don't remember anyone even mentioning tuition. This is especially surprising since the cost of a maester's chain is not trivial-- the metals used to forge one might include silver, gold, and even Valyrian steel.

Does the Citadel in Oldtown charge tuition to educate a student or to grant the title of maester?

No mention is made of a tuition and I doubt there is one

Novices can come from anywhere and any background it seems likely that there isn't a fee. For example, Pate was a baseborn who joined the Citadel.

What I would like to know; -What are the criterions for being allowed into the Citadel for maester education? Could a dwarf in his late twenties like Tyrion be allowed in?

You'll learn more about the Citadel in future books.

In brief, however, there's no age requirement... nor any other sort of requirement, actually. The Citadel is very loosely structured, and open to all. That doesn't mean that all become maesters, however. Oldtown is full of aging novices and acolytes who have never finished forging their chain (or in some cases, never begun)... the Seven Kingdoms equivilent to the eternal grad student of our world.

_{Westeros, So Spake Martin, CITADEL ENTRY REQUIREMENTS}

"A penny will serve. For a silver stag Lorcas will carry you up to the Seneschal on his back. He has been fifty years an acolyte. He hates novices, particularly novices of noble birth."

_{A Feast for Crows, Samwell V}

The Citadel is also funded by Oldtown and Lords who have Maesters in their employment so it might be that they don't need tuition to be able to teach people.

How does the Citadel get financed? Lords pay for the service of the Maesters and the Citadel collects some of the revenue of Oldtown via taxes.

_{Westeros, So Spake Martin, CONJOSE (SAN JOSE, CA; AUGUST 29-SEPTEMBER 2)}

And Acolytes can even be hired to write and read letters for extra funding for the Citadel.

The gates of the Citadel were flanked by a pair of towering green sphinxes with the bodies of lions, the wings of eagles, and the tails of serpents. One had a man's face, one a woman's. Just beyond stood Scribe's Hearth, where Oldtowners came in search of acolytes to write their wills and read their letters. Half a dozen bored scribes sat in open stalls, waiting for some custom. At other stalls books were being bought and sold. Sam stopped at one that offered maps, and looked over a hand-drawn map of Citadel to ascertain the shortest way to the Seneschal's Court.

_{A Feast for Crows, Samwell V}

~It's also worth mentioning that Novices and Acolytes spend years of their lives forging their chains and masters of their subjects.

Alleras would make a maester. He had only been at the Citadel for a year, yet already he had forged three links of his maester's chain. Armen might have more, but each of his had taken him a year to earn. Still, he would make a maester too. Roone and Mollander remained pink-necked novices, but Roone was very young and Mollander preferred drinking to reading.

_{A Feast for Crows, Prologue}

That's a lot of commitment and after all the order is there to serve the realm and they need people from the realm to be able to do as such. Maester's, similarly to the Night's Watch, then swear oath's of celibacy, renouncing family and lands and whatnot so again there is a lot of commitment.

In short as the Citadel is funded from external sources and anyone can join so a tuition fee seems extremely unlikely.

It seems not.

According to So Spake Martin ("Word of God" from GRRM):

How does the Citadel get financed? Lords pay for the service of the Maesters and the Citadel collects some of the revenue of Oldtown via taxes.

If they charged tuition, this would be part of the way they make money, and Martin would presumably have mentioned it. (Unless of course he simply hasn't thought about this issue, and could retcon it later by saying "oh, I didn't mention it then, but they do charge tuition as well".)

It's not completely conlusive, but this may be the best evidence we have.