You would have to work it out with the Game Master
The spell Permanency, from Player's Handbook makes a few selected spells permanent, however there is also a suggestion that other spells could also be made permanent, if Game Master would allow it.
The DM may allow other selected spells to be made permanent. Researching this possible application of a spell costs as much time and money as independently researching the selected spell (see the Dungeon Master’s Guide for details). If the DM has already determined that the application is not possible, the research automatically fails. Note that you never learn what is possible except by the success or failure of your research.
Potentially any spell can be tattooed
First, for your specific case of having Burning Hands tattooed on your hand, note the existing Spell Tattoo which works in a similar way to a scroll and is just a one time use. The price listed for this tattoo is:
A spell tattoo has a market price four times as much as an equivalent scroll.
So for a normal Burning Hands of caster level 1, the market price would be 25x4=100 gp but creating it would cost half this price (50 gp) since Inscribe Magical Tattoo follows the rules for magic item creation.
If you don't want to have this burning hands spell to be single use you can also use the option of creating a new tattoo:
New magical tattoos can be researched and designed using the guidelines for pricing new magic items. Magical tattoos are treated as slotless magical items for pricing purposes.
By consulting the section on pricing magic items, you can figure out how much it would cost to create a tattoo of a spell with more ways to be activated (remembering that as tattoos count as slotless items they would always cost double).
Since no limitation is placed on which spells can be inscribed this way, any spell can potentially be tattooed with DM approval.
Best Answer
Custom Magic Item Creation
Following Magic Item Creation Rules, this would be pretty simple and straight forward.
How I Would Do It (Opinion)
Let's break it down.
Base Cost: From the first entry under Spell Effect (Command word), we can see that the base cost should be Spell level x Caster level x 1,800 gp.
Modifications: For a tattoo like the one you mentioned, it would be a slot-less item, which adds the "No Space Limitation" modification, which doubles the price. And, as you suggested, using X times per day, you divide by (5/X), where the higher value for X results in a higher price. This second modification is taken from the "Charges Per Day" row under Special.