The invisibility rules claim that "An invisible burning torch still gives off light". Does this mean that if I'm invisible and on fire, my position can easily be found by sight alone?
The wording of the spell's description always uses the word "invisible" to describe the spell's effect on its target, and clarifies in its first sentence that becoming invisible means vanishing from sight. It is therefore impossible to see an invisible target, both by the common English definition of the word and by the game's own definition of the term.
The spell's description explicitly says, "Light, however, never becomes invisible, although a source of light can become so (thus, the effect is that of a light with no visible source)." This is where things get tricky.
We, as people raised and educated in the modern day, tend to think of light as something we see: We know that the light reflected or emitted by objects reaches our eyes, and is interpreted by our brains as images. To this way of thinking, the phenomenon described by the quote above seems impossible, as we assume that any object that is invisible must be so because it does not emit or reflect visible light.
However, this is not the only way to think about light. For a long stretch of human history, it was more common to think of light as something you see by: A light source was a phenomenon that, if brought near enough to an object, would cause that object to become visible. The detail that the image of the object is transmitted to our eyes by reflected light was irrelevant and (to a pre-scientific mind) inobvious.
If we use this second way of thinking about light rather than the first, the "light without a source" quote makes much more sense: A lit candle is a source of light, and illuminates its surroundings, and can continue to do so even if we can't see the light source itself, just as if the candle were behind our heads.
I suspect that this second way of thinking about light is probably more relevant to the invisibility spell than the first. Accordingly, an invisible person on fire (and the flames themselves) would be impossible to see, but would still shed light on their surroundings, allowing surrounding objects to be seen.
Of course, this does also imply that an object illuminated by an invisible light source would cast a shadow in the direction opposite that light source. This means that if a flaming invisible person was rolling in the middle of a cluttered room, you might be able to guess their location by looking at the shadows cast by objects.
Being able to guess an invisible creature's approximate location by looking at secondary phenomena does not actually allow us to see the light source itself, however, and (as pointed out by Matthieu M. and KRyan in the comments) is already well-supported by the rules: A character who guesses a five-foot square contains an invisible creature can attempt to attack the creature, but will have a 50% miss chance due to not being able to see their target if it is present in the square, and no chance of all of hitting if the creature isn't there to hit.
As pointed out by WakiNadiVellir's comment, where there's fire, there's smoke. It is unclear whether the smoke emitted by a flaming invisible creature is visible, as there are two lines of the invisibility spell that suggest two different answers.
The first of these lines is "items dropped or put down by an invisible creature become visible." Smoke is mostly composed of fine particulates that were formerly part of the flaming object, so you could argue that smoke consists of millions of tiny items, each of which is "dropped" by the flaming creature as it burns.
The second line is "any part of an item that the subject carries but that extends more than 10 feet from it becomes visible." This line suggests an alternative interpretation, where the "plume of smoke" emitted by a flaming invisible creature is a single item, of which any part that extends more than a certain distance from the creature becomes visible.
On the one hand, the rules often model clouds of smoke and gas as objects or effects that cover an area, rather than as configurations of smaller objects that must be handled individually (see gaseous form, pyrotechnics, obscuring mist, incendiary cloud, et al), and treating smoke as part of the creature and its gear is in line with treating fire as part of the creature and its gear; but on the other hand, the argument that a flaming creature is "carrying" its smoke plume sounds rather strange.
To me, both of these interpretations seem plausible enough to adopt. In the absence of any clear indication as to which is "correct," it is up to each individual GM to decide which they should use in their game.
Personally, were I were GMing, I'd lean towards the second interpretation. This is because when I'm adjudicating a spell's effects, I assume that spells are able to perform their stated function in "normal dungeon-crawling situations." In the case of invisibility, the function of the spell is to render the target invisible, and it's normal for low-level adventurers to carry light sources that emit smoke, so I presume that the spell renders the smoke emitted by an invisible flame non-visible until it gets ten feet away from the creature carrying it. Again, though, that's just my personal interpretation.
According to the rules as written, items picked up by an invisible creature "disappear if tucked into the clothing or pouches worn by the creature." Presumably, therefore, items picked up by an invisible creature that are not subsequently tucked into pouches or clothing remain visible.
I'm not sure whether a fire counts as an "item" or not. My gut says that it doesn't, and that it's instead a quality of the item or creature that's aflame - but if a fire is an item, and a creature catches fire after turning invisible, and catching on fire counts as picking up an item, then the flames will be visible even if the creature is not.
Of course, if they were already on fire when they turned invisible, this caveat is moot; the fire would definitely be invisible.
Probably yes, but ask your DM
Here's the rule from the invisibility spell:
Light, however, never becomes invisible, although a source of light can become so (thus, the effect is that of a light with no visible source).
So it sounds like people can see light coming off the flames outlining your body. If they look in your direction they will see flames suspended in midair.
The DM might rule that you get some level of concealment depending on how much of your body is outlined in flame. But people can tell which 5ft square you are in.
The creature or object touched becomes invisible, vanishing from sight, even from darkvision. If the recipient is a creature carrying gear, that vanishes, too. If you cast the spell on someone else, neither you nor your allies can see the subject, unless you can normally see invisible things or you employ magic to do so.
Items dropped or put down by an invisible creature become visible; items picked up disappear if tucked into the clothing or pouches worn by the creature. Light, however, never becomes invisible, although a source of light can become so (thus, the effect is that of a light with no visible source). Any part of an item that the subject carries but that extends more than 10 feet from it becomes visible.
Fire: is the rapid oxidation of a material in the exothermic chemical process of combustion, releasing heat, light, and various reaction products
The creature or object and the clothes on the creature would be invisible, but FIRE, which is the process of combustion of particles, which are neither being carried, worn, and not tucked under clothing or into a pouch, and are not even directly in contact with the clothes, object or creature, would most certainly be visible. To understand this better, think of flames as little grains of sand that emit light. If an invisible creature pours sand out of a pouch, drops, or puts down the grains of sand, they would become visible, right? The same goes for flames - they are particles, that are effectively being dropped, but because they are lighter than air, they float up (fast), and also emit light.
As such, it's more logical, that an invisible person or item on fire would appear as flames that are burning/floating in mid-air, where the surfaces of the invisible person or item are.
Just because we cannot see the individual particles of flames, does not mean they are not there, and does not make them invisble, in the sense that they are not inherently affected by an invisibility spell The particles are the size of molecules, but particles nonetheless, and since all the individual particles are not affected by the invisibility spell, the flames are visible to us.
The interpretation that particles of fire and smoke should and would be visible, also sits well with the fact that one can do a perception check on the smell of an invisible object or creature - smell is a sense that depends on the ability to detect aromatic particles in the air. Thus, if we are able to detect particles that smell, we are also able to detect particles that emit light (i.e. flames) and particles that reflect light (i.e. smoke)