So, something becomes a black hole when there's too much mass in too small a space.

For a given amount of mass, that's the Schwarzschild radius:

https://en.wikipedia.org/wiki/Schwarzschild_radius

Any object whose radius is smaller than its Schwarzschild radius is called a black hole.

A Dyson sphere would need to avoid collapsing its matter into something smaller than the Schwarzschild radius; if it did, then it would become a black hole. If they don't collapse, then no.

I don't know how Dyson spheres are supposed to avoid gravitational collapse.

goes looking

Okay. Looks like what they do is to basically consist of a bunch of solid satellites that are in orbit but don't collide. They aren't actually a single solid object; the name is something of a misnomer:

https://en.wikipedia.org/wiki/Dyson_sphere

Since Dyson's paper, many variant designs involving an artificial structure or series of structures to encompass a star have been proposed in exploratory engineering or described in science fiction, often under the name "Dyson sphere". Fictional depictions often describe a solid shell of matter enclosing a star – an arrangement considered by Dyson himself to be impossible.

Dyson did not detail how such a system could be constructed, simply referring to it in the paper as a 'shell' or 'biosphere'. He later clarified that he did not have in mind a solid structure, saying: "A solid shell or ring surrounding a star is mechanically impossible. The form of 'biosphere' which I envisaged consists of a loose collection or swarm of objects traveling on independent orbits around the star".[6] Such a concept has often been referred to as a Dyson swarm;[7] however, in 2013, Dyson said that he had come to regret that the concept had been named after him.[8]

https://old.reddit.com/r/AskScienceFiction/comments/zqg6e/is_a_dyson_sphere_actually_possible_or_would_it/

A single-solid-piece dyson sphere isn't possible with any known material; even if it was spinning to neutralize its weight at the equator the poles would still need to support their own weight.

...actually that isn't strictly true only in the sense that we're used to thinking about Dyson Spheres. An extremely light-weight sphere could be supported by the light of its own star, like a solar sail. Relatively small habitats could be periodically hung from the surface, supported by a root system of thin supports spreading into the light-weight sail. (Did I just invent an original setting?)

The common solution isn't a solid sphere, but a "Dyson Swarm"; originally thought of by Freeman Dyson himself. Instead of one solid object it's a myriad of smaller objects orbiting so densely that 100% of the star's energy output is captured and processed before radiating into space.

But a solar-system-sized sphere of gas can't do that, because you can't keep the orbits of the gas from smacking into each other.