The main barrier here is the fluorine resistance; once fluorine has bonded to something, chlorine or oxygen will not replace it! However, if the creature's tissues aren't already fluorinated, they will rapidly become so in a fluorinating atmosphere! So, if the creature's tissues must be naturally fluorinated, how could it possibly survive on anything other than fluorine?
They key is to recognize that it can't be fully fluorinated, just as we aren't fully oxidized! Whatever the details of this fluorinated biochemistry are, there must be unsaturated bonding sites which could be broken up for energy and occupied by oxygen or chlorine--otherwise, they'd just be a cloud of CFl4 or SiF4 gas or something like that, rather than an organism made up of complex biomolecules. And while fluorocarbons are famously not flammable in air, that doesn't mean they won't exothermically react with oxygen at all--just that the activation energy is too high for the reaction to be self-sustaining. Plenty of fluorocarbons will, however, combust in higher concentrations of oxygen--and overcoming activation energy is what enzymes are all about!
Fluorocarbon liquids are also famous for their ability to dissolve gasses. They are, in fact, used for liquid breathing, because they can dissolve sufficient oxygen for respiration directly. Thus, given perfluorocarbon-based blood, these creatures would not need to contend with developing reversibly-binding hemoglobin-equivalents for all three possible oxidizers as was brought up in Sean OConnor's answer.
So, for simplicity, assume the creatures are still carbon-based. Their biomolecules will have a lot of fluorine atoms substituting where we would use terminal hydrogens or -OH groups, such that they are not corroded by fluorine gas. Being resistant to fluorine, they are also unbothered by the chemical reactivity of oxygen and chlorine. Chlorine might be a metabolic poison, in the way that, e.g., carbon monoxide is a poison for us (because it displaces oxygen but is not oxygen), or arsenic (pretends to be phosphorus) or strontium (pretends to be calcium)--but we'll suppose that they have instead evolved more specific enzymes that can discriminate between chlorine and fluorine. Their body tissues, being perfused with perfluorocarbon liquids, directly dissolve large quantities of oxygen, fluorine, and chlorine from the air (along with other gasses), so all three are available to respiration, so long as they are available in the environment.
When they have access to fluorine gas, that's the preferred oxidizer, and the creatures would end up exhaling mostly CF4 and HF, with some COF2 (assuming that oxygen is still used as a frequent heteroatom in their biochemistry--and why wouldn't it be?), possibly some free oxygen (OF2 production is not thermodynamically favorable at normal pressures), and NF3 as a result of protein metabolism; no need for these guys to produce urea!
Given access to oxygen but no fluorine, they would fall back on a less-energetic system, producing almost exclusively COF2 (the result of reaction -CF2- monomers with oxygen), and probably some water. Terminal trifluormethyl groups would be unnattackable, but those could be packaged and eliminated as hexafluorethane. Protein metabolism could just release gaseous nitrogen directly--high gas solubility means that they still have no need to package waste nitrogen as urea!
Finally, chlorine metabolism would be the last resort, producing the least energy, but still more than anaerobic respiration / fermentation. Perfluorocarbon chains would produce primarily CF3Cl (Freon-13) and CF2Cl2 (Freon-12) waste gasses, along with COCl2 (phosgene)--there would be no free oxygen produced, because chlorine (unlike fluorine) won't displace oxygen, so it would just fill the extra slots of carbons that already have C-O bonds. And just like OF2, OCl2 is not thermodynamically favorable at low pressure.
Given that they can breathe fluorine, there's probably some hydrofluoric acid replacing water, and integration of organosilicons, but that doesn't change the general structure; the non-oxygenated carbon compounds produced as metabolic waste have direct silicon analogs (SiF4, SiF3Cl, SiF2Cl2), which are also conveniently gaseous. Silicon oxyfluorides and oxychlorides are more complicated, so Si-O bonds might be isolated from fluorine and chlorine metabolism to form silica instead, which can be dissolved in hydrofluoric acid for liquid elimination.
