In the books, at least, Floo-calls were green, not red. And the magic of the floo makes the flames warm but not hot (like the Flame-Freezing charm). So no, I doubt the flames would have burned a fully-qualified witch like Molly Weasley. It was, as JdeBP said, more out of politeness and etiquette.

@Jeff - Except that she takes it with her hands, then uses the tongs to put it into the fire.

I, for one, would quite like to see what happens if she sticks her hand in fire! :P

the toast may be warmer than the floo flames :) also, it can still be an issue of etiquette. She cant serve it on a plate, Digory has no access to his hands and she is uncomfortable with just stuffing the toast in his face. Using a tool is the best option, so Molly takes the tongs, which are the most convenient in this case.

@Lt.Ortega - A British person (and certainly one of Molly's social class when speaking to a family friend) would simply say "excuse fingers". The implication being that if the food is clean enough to be eaten, it's clean enough to be passed with the hands.

it strikes me as a tad too intimate a gesture. Molly was always about manners, so I think it's in character for her to use tools and not her hands when serving food. Also, not sure they are close friends with the Diggorys...and in the wizarding community the rules might be different that in the muggle middle class. more old-fashioned, like in @JdeBP's answer.

@Lt.Ortega - They're close enough that he turns up un-announced in their family kitchen and accepts food from Molly

@Valorum I'm British and I don't recall hearing the phrase "excuse fingers" in this sort of context.

@Randal'Thor - What would you expect someone to say when passing you a bun with their hands because there's no tongs?

Floo fire is described as being emerald green in colour, so the flames in the Weasley fireplace are probably not Floo flames but regular fire (against which Diggory is presumably rendered temporarily immune while Flooing). As @Jeff points out and A History of Magic makes clear, though, even red hot flames would not have burned a fully-qualified witch if she didn’t want them to. So the flames are likely not the reason. Nor is etiquette or awkward positioning. Which leaves us with… no good reason at all that I can think of.

@JanusBahsJacquet - Before she puts something into the fire she has the option of using magic to make herself immune to flames (a spell which presumably requires her to fetch and then use her wand, presumably along with some mental preparation) or she could just use the damn tongs that are right in front of her.

@Valorum And which are grimy, dirty, and covered in ash and soot from being used to stoke fires for years. Do we ever see Molly not have her wand at the ready, even in the house? There’s nothing in any of the books to support your presuppositions; it’s just as likely that all she’d have to do was take her wand out of her housecoat pocket and perform a simple, easy spell.

@JanusBahsJacquet - Molly isn't opposed to using magic for drudgery, but she also seems to oppose using magic frivolously.For all we know, the "don't get burned" spell is moderately involved or requires some level of preparation

@Valorum Apart from being annoyed at the twins Apparating everywhere (understandable, since they make her jump every time they do it) and strict about underage wizards not being allowed to use magic outside school, where does she give that impression? She uses magic for simple tasks all the time; a Flame-Freezing Charm doesn’t seem like anything that she would have any kind of trouble with.

@JanusBahsJacquet - It strikes me as overkill for passing someone a piece of toast.

@Valorum No more so than using whatever other spells she was using to make a dustpan scoop up potato peels or to open a kitchen drawer to get knives. Less so, in fact, since the non-magic alternative there is just taking three extra steps and holding something in your hand; here the non-magic alternative (as it actually played out) was actually quite gross, not to mention quite rude to Mr Diggory.

The instrument for stoking a fire is a poker, not tongs.

But this implies the toast would get burned (possibly even set aflame!) in the process of handing it. How could it be safely put into his mouth if her hand would be burned in the process of handing it over?

@jpmc26 - I'm reasonably certain that it'll be gone before it can heat up enough. Her hand, however, would be in the fire for slightly longer. Why do you normally use tongs to move the logs into the fire?

@JanusBahsJacquet Are we sure a Flame-Freezing Charm wouldn't have disabled communications? Presumably there's a reason people use flames for this sort of thing.

@JanusBahsJacquet - regarding the color of the flames, I always chalked that up as a mistake in the movie. That was absolutely a floo call, and in the books those have always been seen to make the flame turn green on both ends.

@Jeff The only other place I can think of where we see an incoming Floo connection to a grate with an already lit fire is when the Trio talk to Sirius in the Gryffindor common room (are there others?), and the books don’t note the colour of the flames in either case—including any mention that the characters find anything unusual about it. That doesn’t say anything for sure, of course, but it does make it at least possible that Floo flames are green, but a Floo connection doesn’t turn already lit, real flames green.

(When Harry spots Sirius in the fireplace, he doubts what he saw, thinking perhaps it was a trick of the light. Sirius also describes a girl he thinks saw her, but who didn’t seem to think anything more of it. I doubt that would be the case if the flames had suddenly turned emerald green and then back to red—the kids would have noticed something odd about that.)

@Jeff Of course they wouldn't have burned her. Because she would use tongs. As for politeness and etiquette, I must have missed all the stuff on tongs usage.