The vines created by Entangle are flammable because regular vines are flammable
There are no secret rules.
D&D is written using natural language. Unless the game defines a particular term differently we are expected to use the natural language meaning of the word.
It uses natural language so we can use our own experience and understanding of the words in our world as a baseline, with the game providing exceptions to this understanding (like the existence of magic and monsters).
The Player's Handbook starts off the spellcasting section with this statement:
A spell is a discrete magical effect, a single shaping of the magical energies [...] into a specific, limited expression.
In other words, spells produce a limited magical effect. That limited magical effect is specific, and is described precisely by the spell description.
The Entangle spell creates:
Grasping weeds and vines sprout from the ground in a 20-foot square starting from a point within range.
The game (and the spell) do not make any further definition of weeds and vines, and thus we must use the natural language definition of weeds and vines.
Both weeds and vines can be cut, and thus slashing damage at a minimum should be able to damage the vines and thus be used to free oneself.
Similarly, being compromised of plant matter, weeds and vines are both flammable (see real word forest fires), and thus a well placed fireball should set them on fire (but that would also likely damage the creatures in the Entangle spell the fireball was attempting to free).
But grease isn't flammable...
Under the natural language meaning of grease, the substances which make up grease in our word are not universally flammable. As a result the grease created by the grease spell is not flammable by default.
There is, however leeway, for a DM, to rule that grease created by the Grease spell in their world is flammable, and still be following RAW.
Isn't this just another "secret rule"?
Let us consider the inverse situation:
The vines are not flammable and are impervious to all damage because the spell doesn't say they can be damaged.
This interpretation relies on the secret rule that things aren't flammable or capable of being damaged unless they say they are.
Let's take this interpretation to the extreme. Let's say I have a spell that creates dry sticks, Create Dry Sticks. This full text of the spell is:
A cubic foot of dry sticks are created at your feet.
As the spell does not say that the sticks are flammable, they aren't flammable. Indeed, it doesn't say they can be damaged in any way.
As a result, we can't use these sticks to make a fire with. Indeed...we have created indestructible sticks! Bundle them together and we now have an indestructible weapon (made of dry sticks), or perfectly resistant armour (made of dry sticks).