Dwarven constitución & multiclass

if I have a dwarf character that is multiclassing and has the feat dwarven fortitude. Which class hit die should I choose?

That would be your choice.

Hit Die are a finite resource similar to spell slots, you gain a single die per level, the size of the die depends on the class in which you gain said level. When you spend a die, either during a rest to regain HP or through some other mechanic, unless otherwise specified, you can choose which die you wish to spend to do so. Once you have spent a hit die, you can no longer use it until it has been recovered, you recover up to half of your maximum hit die per long rest. 

As an example a Lvl 4 multiclassed character, 3 Levels in Cleric, 1 level in Paladin would have a hit die pool of 3d8 + 1d10. In battle you could choose to spend 1d10 to recover HP using your feat, this would leave you with 3d8 + 0d10 available hit die to spend later either during a short rest or using your feat once more. If you used a further 2d8 through the day by the time you reached a short rest you would have a hit die pool of 1d8 + 0d10. During a long rest you can recover up to half your total hit die, in a combination of your choice, allowing you to recover 1d10 + 1d8 or 2d8 still leaving your 1 Hit Die short of your maximum. 

You pick from your available hit die and roll, just as if you were rolling a hit die during a short rest. And then mark that hit die off as used until it's recovered by a long rest. 

Dwarven Fortitude lets you use your hit dice (one at a time) when you take the Dodge action as if you were taking a short rest. You have one hit die per level, so if you're a second level fighter and first level rogue, you have 2d10 and 1d8 to choose from each time. If you use a fighter HD with Dwarven Fortitude, you roll a d10, add your Con modifier, and regain that many HP. Then you have 1d10 and 1d8 left to choose from the next time you use Dwarven Fortitude or take a short rest. You regain hit dice as normal when you take a long rest. 

Short answer: Dwarven Fortitude lets you compress part of a short rest into combat.

This thread has seen very informative for me and I've learned a few things. However, I think people have missed a key word - you've asked which you should use, rather than what you can use.

It's personal choice and it is highly situational. How desperate are you to get to full HP with as few Hit Die uses as possible? You may want to risk "wasting" HP if you really need to be at as high HP as possible. Also, if you're almost at the end of the day and you're sure that you're going to be having a long rest afterwards, you may not be bothered about wasting a large Hit Dice versus a small one. In both cases, I'd tend towards the largest.

However, as a rule of thumb, I'd compare the max HP that you could potentially recover from the HD and then compare that to how much HP I'd have to recover to regain max HP. I'd use the largest Hit Dice which has the max recoverable HP that isn't larger than would be required to Hit max HP. So, for example:

Let's say that you've done a three way multiclass between a Wizard, a Druid and a Paladin, giving you the choice of a 6d, 8d and 10d Hit Dice, and have a Con of +1. You're in a battle and lose 10HP.

- The 6d could give you a maximum of 7HP, you'd certainly not waste any, but won't fully heal you.

- The 8d could give you a maximum of 9HP. You wouldn't waste any, but it will still leave you 1HP down.

- The 10d could potentially heal you 11HP. This could fully heal you, but could waste 1HP.

To prevent wastage, I'd us the 8d, because it offers you the most healing without wasting any. However, it is highly situational and down to personal taste. If I'm feeling under threat, I'd take the d10 because it's more likely to give me more HP, and that's worth the risk of wasting HP. Extra HP on your Hit Dice are worth nothing if you're dead.

Thank you all very much, I think, my doubts have been clarified.

Quote from Linklite >>

This thread has seen very informative for me and I've learned a few things. However, I think people have missed a key word - you've asked which you should use, rather than what you can use.

It's personal choice and it is highly situational. How desperate are you to get to full HP with as few Hit Die uses as possible? You may want to risk "wasting" HP if you really need to be at as high HP as possible. Also, if you're almost at the end of the day and you're sure that you're going to be having a long rest afterwards, you may not be bothered about wasting a large Hit Dice versus a small one. In both cases, I'd tend towards the largest.

However, as a rule of thumb, I'd compare the max HP that you could potentially recover from the HD and then compare that to how much HP I'd have to recover to regain max HP. I'd use the largest Hit Dice which has the max recoverable HP that isn't larger than would be required to Hit max HP. So, for example:

Let's say that you've done a three way multiclass between a Wizard, a Druid and a Paladin, giving you the choice of a 6d, 8d and 10d Hit Dice, and have a Con of +1. You're in a battle and lose 10HP.

- The 6d could give you a maximum of 7HP, you'd certainly not waste any, but won't fully heal you.

- The 8d could give you a maximum of 9HP. You wouldn't waste any, but it will still leave you 1HP down.

- The 10d could potentially heal you 11HP. This could fully heal you, but could waste 1HP.

To prevent wastage, I'd us the 8d, because it offers you the most healing without wasting any. However, it is highly situational and down to personal taste. If I'm feeling under threat, I'd take the d10 because it's more likely to give me more HP, and that's worth the risk of wasting HP. Extra HP on your Hit Dice are worth nothing if you're dead.

 We had one of the situations where overflow might be acceptable last night. We were facing sahuagin, who have an ability that grants them advantage on attacks against anyone with less than their max hit points. I usually hate to waste healing, but in that case it was much more important than usual to be fully topped off.