It depends on the illusion
Blindsight says:
A monster with blindsight can perceive its surroundings without relying on sight, within a specific radius.
Creatures without eyes, such as grimlocks and gray oozes, typically have this special sense, as do creatures with echolocation or heightened senses, such as bats and true dragons.
So, the question is if the illusion fools the other senses the creature uses instead of sight to perceive its surroundings.
While illusions in general are of magic that dazzles the senses, befuddles the mind1, the various illusion spells differ, and will tell you what senses they fool.
Illusions that do not work against blindsight
These are mostly illusions that are purely visual, so that blindsight does not interact with them and does not perceive them at all.
You create a sound or an image of an object within range that lasts for the duration.
Here, you can only create either a sound or an image (i.e. a visual illusion), so this would fool either hearing or normal sight. Blindsight would not be fooled by a mere insubstantial and silent image, as if you do not use sight to perceive your surroundings, you would not register that image. A bat's echolocation would go right through this image.
- Or consider the illusion created by the spell silent image, which says:
You create the image of an object, a creature, or some other visible phenomenon [...] The image is purely visual; it isn't accompanied by sound, smell, or other sensory effects.
Since this image is purely visual, it will not fool blindsight, which does not perceive visual things.
- Even more on the nose, the illusion mirror image outright tells you that blindsight will work to overcome it
A creature is unaffected by this spell if it can't see, if it relies on senses other than sight, such as blindsight, or if it can perceive illusions as false, as with truesight.
One could use this as an argument that if this spell needs to explicitly say so, any spell that is overcome by blindisght or truesight must say so, but I think that is putting too much on this helper text. It would have been sufficient to state that the illusion is purely visual; as this is one of the spells that is most likely to see combat application, I think the text is there to directly help address questions that come up.
Illusions that fool blindsight
Compare this to illusions that would fool blindsight. These are illusions that work against all senses, that create something that can be perceived by all senses, or that bypass the senses and are directly created in the vicitims mind:
- Some illusions work no matter what sense you use. The Invoke Duplicity option for Channel Divinity of the Trickster cleric domain says
As an action, you create a perfect illusion of yourself that lasts for 1 minute
A perfect illusion will fool all the senses. It does not matter if you have an alternate way to perceive. This illusion will fool echolocation and blindsight.
- Some illusion spells create quasi-real things that can be perceived and will fool blindsight. One example is the spell phantom steed which is also an illusion school spell and says:
A Large quasi-real, horselike creature appears on the ground in an unoccupied space of your choice within range [...] The creature uses the statistics for a riding horse
The creature does have substance even though it is an illusion. You can even ride on it. Echolocation will bounce of of it and creatures with blindsight will be able to perceive it.
- Lastly some illusions also create things in the victims mind, often called phantasms. These will normally allow the creature a saving throw. One such example is the phantasm created by the spell phantasmal force
You craft an illusion that takes root in the mind of a creature that you can see within range. [...] On a failed save, you create a phantasmal object, creature, or other visible phenomenon of your choice [...] The phantasm includes sound, temperature, and other stimuli, also evident only to the creature.
If they are successful these will always fool, as they are not perceived via the senses.
1 The inset box for The Schools of Magic (PHB, p. 203) describes the school of Illusion as follows:
Illusion spells deceive the senses or minds of others. They
cause people to see things that are not there, to miss things
that are there, to hear phantom noises, or to remember
things that never happened. Some illusions create phantom
images that any creature can see, but the most insidious
illusions plant an image directly in the mind of a creature.
So, some illusions only create images, others don't only create images, as we also can see by the examples given.