Is there any evidence, in the rules or from developer comments, that parts of the description of a D&D 5e spell is “only flavor text” and should be ignored for determining what happens in-game?
Is there evidence to the contrary?
There is certainly flavourful text in spell descriptions, but spell descriptions are rules so it's all rules text too.
In an answer to Does Burning Hands really require touching thumbs? I eventually wrote a detailed argument to support the general concept that spells descriptions are rules — i.e., that there is no such thing in D&D 5e as spell text that is only ignorable “fluff” or “flavour text” — in order to respond to the comments objecting to the original, quite short answer. Writing that general essay to support the specific yes/no answer about that one spell seems prescient now, since it answers this general question directly; so I will quote myself:
Spell descriptions are rules[.]
The alternative—selectively designating some parts of the spell as "real" rules and other parts as not—is entirely doable as a DM's house rule, but as it requires a personal interpretation it has no foundation in the rules as written beyond the permission they give to the DM to create house rules.
Addressing the objections
One common motivation for assuming that there must be a distinction between "ignorable fluff" and the "real rules" is familiarity with game designs that deliberately create such a distinction. That is an idea that isn't part of most editions of D&D, including what we've seen of 5e's core. Such a separation is a prominent feature of 4e, and that might lead one to believe that it naturally exists in all D&Ds or even all RPGs, but separation of game text into "fluff" and "crunch" isn't the default in RPG design, and is particular to the RPGs that feature it.
It's plain that no such distinction exists overtly in 5e spell descriptions: they have no separate "flavour text" or any other distinguishable separation between required and optional parts that can be pointed to and agreed upon, without making an arbitrary decision. All of the descriptions are in terms of what happens, freely mixing mechanical and the fictional effects with no demarcation to say that this is a rule but that can be disregarded. Expecting such a separation, taking it as a given, will motivate one to look for one, but expecting something to be true doesn't make it so.
It has been suggested that the first sentence is the fluff and the rest of the spell crunch. This argument could be discarded swiftly by pointing out that there is nothing in 5e that ever suggests this, and it's invalid to try to interpret 5e based on the conventions and structures of some other game; but this dismissal has more support than that. Looking at three spells chosen randomly, it becomes evident that this proposed division doesn't exist: astral projection has rules for targets and duration in its first sentence; resistance specifies a hard condition for its target ("willing") similarly; ice storm's first sentence is the only place its area of effect is given. Indeed, the first sentence of burning hands is the only place where its area of effect is described as a two-dimensional cone rather than a three-dimensional one, a small but substantial rule that could mean the difference between life and death for friend or foe alike. This proposed line between the first sentence and the rest of the spell description as being the fluff/crunch divide has no merits to raise it above any other arbitrary dividing line, and actually has severe drawbacks for being able to use many spells.
An argument could be marshalled that D&D's recent editions establishes a weight of historical precedent for a fluff/crunch divide that 5e is natural heir to. But if that was a reasonable analysis of 5e's text, it would be present unambiguously in the text. That it is difficult to read such a division into the text despite that prominent heritage is a very strong argument that it doesn't exist in 5e. Further, there is actually a much longer history of the indivisible fusion of fiction and mechanics in D&D. This is the heritage that the designers of 5e have been explicit about drawing from in its design, which makes the argument that a division doesn't exist stronger by establishing the likelihood that it's a deliberate design choice. It's hardly something they could have intended to include, but forgot.
Again, all this can be overridden by house rulings of course, but such a separation is not in the rules or even implied, and in many places strongly contraindicated. We may see such guidance in the DMG as an option, but from a structural analysis of the rules, it is not the base state of affairs—the rules resist attempts to locate a dividing line, as questions like this attest.
TL;DR: Spell descriptions are rules in D&D 5e, and you can tell that's the case because the alternative of selectively ignoring parts of spell descriptions is not supported anywhere in the text, and because attempting to apply the concept to the game anyway breaks most spells.
No there's no flavor text in spells. The block of text is called the "spell effect" and the PH states it has a direct impact on game play.
Each spell description in chapter 11 begins with a block of information, including the spell's name, level, school of magic, casting time, range, components, and duration. The rest of a spell entry describes the spell's effect.
(PH p 201, Chapter 10: Spellcasting, under "Casting a Spell")
The physical manifestation of the spell (the part which might sound like flavor text) is specifically called out as relevant:
Unless a spell has a perceptible effect, a creature might not know it was targeted by a spell at all. An effect like crackling lightning is obvious, but a more subtle spell effect, such as an attempt to read a creature's thoughts, typically goes unnoticed, unless the spell says otherwise.
(PH p 204, Chapter 10: Spellcasting, under "Targets")
There is no overarching statements about flavor text, but several spells have open opportunities for "flavorful" interpretation. Planar Ally, for example, leaves wide open to DM and player negotiation/interpretation exactly who can be summoned and the form of payment. The fact that such example exist suggest that the game designers want DMs to feel free to flesh out spell details to get the tone they want in their game. The fact that such examples are rare suggest that they want DMs to be judicious in doing so, perhaps because flavor text can sometimes be exploited by creative players to create unbalanced effects.