It is possible to make a party with Fina for auto-repeat farming Faith, but it will depend from player to player and the units being raised.
To do this we need to understand how her Cheer ability and AI works:
The Cheer Ability
Cheer raises MAG/ATK and FAITH for 3 turns, this means that your party needs to finish the battle before the buff runs out so that 25% of the buff effect is carried over permanently at the end of the battle.
How much Faith is raised depends on the ability level, an ability level of 12 is recommended, before that the Faith values are so low that you will not see a full point of faith being raised in the character stats menu after battle. However, Faith is being raised, but by small not visible values (50 to 50.4 Faith will still show 50 in the menu).
After that comes Fina’s AI, usually she will use Cheer on herself first, then on the rest of the party. This isn’t random, she will cast it on the first in range party member in the formation from left to right so pay attention to your formation positions before battle.
However, her AI changes if someone in the party has HATE, this change her entire behaviour and target the unit with Hate FIRST. This makes it much easier to auto-repeat, equip a vision card like Vow of love and now you can put units in whatever position you like and as long as Fina gets a turn, she will cast Cheer on the character you chose to put Hate on.
Auto Farming Process
Make a faith farming party with Fina (must complete battle within 3 turns)
Equip Hate Vision Card on the unit you want to raise faith
Test the map (auto once and make sure Fina is casting Cheer and battle is ending in 3 turns) a. If this isn’t happening go back to step 1 and play around with your party.
Put it on auto-repeat, enjoy.
Do not forget to level Fina’s Cheer, you will get a lot of JP on her with this method, the more you level that skill, the faster you raise faith.
Choose a fast map, Chapter 1 Section 3 Battle 3: False Accusations for example but be careful because AOE units like Mediena can one shot the map so place her in a position where that only happens in the second turn.
Watch out for AGI differences between your Fina and the rest of the party, if your Fina is underleveled you may want to remove espers to reduce AGI from other party members.
Don’t be afraid to have big party, with HATE you will get consistent first turn results so play around with the formation.
Remember that even if you don’t have a HATE card you can still do it, you just have some formation restrictions and might need a longer slower battle.
I'm not sure that Faith is raised only for 3 turns. I've got my Cheer to Lv20 and I just run the auto with VoL onto the unit that I want to increase Faith and even if it doesn't finish within 3 turns, the increased Faith remains. However, the AI likes to run Cheer onto the VoL wearer, onto itself, and then whomever's closest after Fina casts it onto herself. Being there's only 3 procs possible in one run, that about sums up the folks whose AP will have been increased for the specific run.
The lingo has "Raises Faith" separated from "and raises ATK/MAG for 3 turns." I'm pretty certain that Cheer is a permanent increase of Faith in the particular stage. But only 1/4 of the increased Faith is permanent. I may be wrong, though... but the language suggests that even if there was no increase in ATK/MAG, there is still an increase in Faith.
I thought the "make sure it only takes 3 turns" was because Fina can only cast Cheer 3 times.
You need a minimum skill lvl of 10 on cheer for the Faith effect to increase after battle. If she's on auto she won't target the same unit, but if you manually cast it you can hit the same unit 3 times at lvl 10 for 3 skill ups per battle.