Some chants might sound tonal-ish, others not. It depends on the mode and the features of the particular melody. I find this stuff interesting and fun to talk about, so here goes!
In Gregorian chant there were eight modes, each defined by a range and a final (ending note). The four "authentic" modes were basically defined by white-note scales starting on their finals: Dorian (D to D), Phrygian (E to E), Lydian (F to F), and Mixolydian (G to G). Sometimes they also went one note below the final. (I'm using letters corresponding to white keys on a piano for convenience, but in the middle ages these weren't absolute pitches - it would be more accurate to say Dorian ran from re to re, Phrygian from mi to mi, etc.)
Each authentic mode had a corresponding "plagal" mode with the same final but with a range running from a fourth below the final to a fifth above; these were called Hypodorian, Hypophrygian, etc.
In the Dorian, Lydian, Hypodorian, and Hypolydian modes, B flat was sometimes used in place of B natural. This means that in practice they were often more like our major and minor modes. Later theorists would recognize two other modes, the Ionian (C-C, major) and Aeolian (A-A, minor), but the medieval theorists didn't.
Each mode also had a reciting tone, which was a melodic point of emphasis, usually sounding more often than the final.
An important thing to note is that the medieval chant composers didn't like the sound of the leading tone cadence that basically defines tonal music (i.e., E-->F) and consciously avoided it. That certainly weakens any sense of tonality in Gregorian chant.
So what does this mean for how it sounds, whether it has tonality?
Here's a seasonally appropriate chant in the Hypophrygian mode or mode 4 (medieval theorists mostly used numbers):
To most modern listeners unused to Gregorian chant this will sound like it ends on the wrong note. The melody and in particular the reciting tone of this mode, A, suggest to us a home of D, but in the medieval system it ends on its proper final, E.
Here's another beautiful and relatively well-known chant, though not seasonally appropriate:
This one is in mode 1, Dorian. Note the B flats, except the B natural descending line in the middle - that kind of switching between B flat and B natural is characteristic of the medieval Dorian mode.
To me, this one sounds more tonal. It sort of sounds like it's in F but then veers to the relative minor D for the final alleluia. But again, within the medieval modal system, it makes perfect sense.
And here's the most famous chant melody, the Dies Irae:
Straightforward Dorian melody here - pretty easy for our ears to understand and fit with our tonal expectations, IMO. This one was written later, 13th century, and it has a big range, going down to the A below the final...earlier chants in mode 1 wouldn't do this.
You asked about Lydian (mode 5) specifically. Here's an example I found:
Notably, it uses B flat a lot. We might say it sort of vacillates between major and Lydian...or maybe between F major and A minor.
When Beethoven and Bruckner decided to use the Lydian mode for a special effect, they stuck to the Lydian scale strictly and avoided B flat. But medieval and Renaissance composers never really did that.