There are already questions like this (creature which set itself on fire for defense) and this (creature with flaming collar), asking about flaming creatures or creatures which ignite their bodies with different reasons also we have a lot of questions asking about dragons, fire breathing or fire manipulation, at each one are explained how the ignition mechanisms work, so gonna forget how those all creatures produce they fire and let's focus in how they survive their own fire (unless the way in which produce their fire could influence in this topic).
First with the respiratory problems, producing fire gives other unwanted by-products as fumes, water vapor, ashes and principally carbon dioxide and carbon monoxide, this latter being the principal problem: CO, is an extremely dangerous molecule to any living organism and till where I found there's no way to form resistance, cure or block the CO intoxication.
So the solution would be use blue fire based on different reagents permitting a complete combustion without the CO as by-product (I think that here the way of producing fire influences because if normal fire is difficult to justify, full combustion fire is even more difficult), but here is the problem that the required oxygen to produce the combustion is the same that the creature needs to breath, in relative constant use of fire as biological mechanism, practically its own fire will steal its air.
Maybe the solution could be hold the breath, but depending on the time and size this require excessive big lungs or reduce the metabolism which as a consequence reduces moving: just imagine using fire during a fight and have to stay still administering the oxygen, plus the own fire will make necessary breathing for reduce the internal temperature, while practically it is needed to fast moves to get away from the burns like fire dancers do, even having a higher resistance is inside the biological limits.
Circulatory problems are less again practically just related with oxygen transport and problems with CO, but other is the blood warming, at the dragon's question an answer mentioned that wings could be used as cooler due to their high vascularization, but that is a double-edged sword, if you start to surround yourself with fire your wings will increase overheating and if you do not have wings or have fire in one part of your body it is even worse. Also is very probable that a gigantothermic or endothermic creature will suffer heat easily than an exothermic creature.
And finally there are other problems like that its eyes will burn caused by smoke and dry out due to intense heat or having to adjust to the limits of its own fire, but I just mentioned these two systems because I consider them more related with each other.
So my questions is, how the respiratory and circulatory systems could "evolve" or be modified in flaming creature or fire-breather to resist its own fire and by-products?
(Go to this question for check what is a flaming collar creature)