"Is there a best or does it depend?" Depends.
"And if it depends, what does it depend on?"
Particle board= Inexpensive,smoothest ,square and flat but suffers most if exposed to water.
Plywood= Modest cost, supports heavy load.smooth,square and reasonably level. Suffers little when exposed to water.
Solid wood constructed of glued up narrow boards= Most expensive, minor defects in flat,level and effected least when exposed to water.
Most important consideration = which material maker preferrs working with. Most often overlooked certainty= veneered pieces are least passed from generation to generation.

Thanks for this. What I'm guessing is true, in terms of passed from generation to generation, is that veneers are likely to be less 'timeless' than non-veneers. That is, with a pattern or something, it's going to be subject to changing styles. Or is it something about quality?

Are there any issues with chemical release (once the table in done) from either particle board or fiber board?

The veneer will remain adheared for years unless something soaks it loose or swells it.
Veneered funiture is discarded primarly because it's generaly thought of as lower quility than soilid. Is next to impossible to restore once painted. Challenging to repair chips,scrapes and gouges, looks worse when chiped and looks even worse with each additional chip. Many view battered and worn wood as having "caracter"but few will say the same for veneer.

OTOH, most of the furniture that makes it to museums is veneered, often quite spectacularly. Most classic furniture pieces use a combination of solid wood and veneer.

A lot comes down to style. If you're making a piecrust table with elaborate radially bookmatched burl you'd better think veneer; if you're making an English Arts and Crafts table, veneer would be the antithesis of the style.

If I were making a veneered table top I'd probably use baltic birch plywood underneath, and be sure to veneer both sides to avoid warping, and a solid wood edge to absorb abuse. Alternately, MDF with a solid wood edge, again applying veneer to both sides. MDF is the more stable substrate for veneer but makes furniture incredibly heavy.

Rwiegand correctly notes that almost all the high end furnishings preserved in museums have veneers. Most people associate veneer with cheap furniture simply because for the middle class shopping from the big box stores, that's often the case. But historically, the nobility and the wealthy have favored decorative, artistic pieces that used veneer.

As far as repairs go, veneered furniture is repairable but requires more skill to accomplish. Store bought furniture is usually pitched because skilled labor can cost more than new furniture but finer pieces like those in museums will be restored because people value them so highly.

If the OP hasn't already done so, it's probably best to defer to the furniture maker who is building the table. Choice of substrate is a design call that needs to be based upon the circumstance of each build.

As far as off-gassing being an issue, any manufactured sheet materials such as MDF and particle board and even plywood can release small amounts of chemicals over time. If this is a concern, I'd recommend requesting formaldehyde-free materials. They're available in many locations but are a bit more expensive.