[RPG] Are there any alignment restrictions for Zhentarim members in Adventurer’s League

I think the key issue here is that you're thinking of it as "the old system with alignments removed" rather than "the old system simplified." Character archetypes haven't gone away, there's just more variety within each alignment.

Your examples are pretty easy:

• Personified forces of nature (chaotic neutral), aka Greek mythology, which, while dangerous, are not necessarily evil?

  Neither particularly good, nor particularly evil? Sounds unaligned to me.

• The honorable lawful evil villains, who offer their opponent a fair chance in a duel?

  A villain, not driven by Chaos for the sake of chaos? Probably evil, although perhaps drifting into unaligned (that Lawful Evil/Lawful Neutral border has always been a soft one).

• What about the structure of the Planescape universe?

  If you wanted to port these over whole-cloth, you'd need to separate character-sheet alignment from planar alignment. Characters from a Chaotic Good Planescape plane would have a character-sheet alignment of good, but would have personalities different from characters from a Neutral Good Planescape plane.

• The endless war of the Tanar'ri and Baatezu?

  Two groups of people locked in a conflict of ideals. One beholden to the principle of Law, the other Chaos. Both evil. From an alignment perspective they'd be evil and chaotic evil.

Consequences

The long-and-short of it is that the mechanics of roleplaying in D&D have been simplified and softened. The guidance to the DM is to award experience for good roleplaying, but "good roleplaying" is no longer as well defined.

Some of the results of this are:

• Players who create a character, and then assign an alignment to it will have more freedom to create deeper, more nuanced characters, because they have more room to add interesting contradictions within each alignment.

• Players who pick an alignment, and then create a character around it will be more homogeneous, because there are fewer starting points to work from.

• The DM has fewer "sticks" with which to punish players for loose roleplaying. There was an article not long ago that talked about how power had slowly been shifting from the DM to the players in recent editions of D&D, and this is likely part of that. (If someone can psychically deduce what article I'm thinking of, I'd love to link it here).

Opportunities

Moral dilemmas where there is no obvious good and evil choice?

I'm not quite sure how the alignment system impacts this. This seems to be the sort of thing that's always been squarely in the DM's court as a writer. Certainly, the good alignment is enough to get players into plenty of conundrums on its own.

Possibilities for intrigue and conflict when a lawful neutral inquisitor-type character has to cooperate with a chaotic good one, to defeat a great evil.

Those possibilities still exist, the ball is just further in the players' court. The inquisitor might be lawful good or even unaligned, while the other character is simply good, but well-characterized characters will still find reasons to butt heads.

Of course, it still takes good roleplayers to make sure this sort of conflict doesn't go sour!

Both of your examples have been implemented time and again by DMs in systems outside of D&D. The alignment system, while interesting, has proven to be non-vital to the process of roleplaying.

From Monster Entry Format

Alignment, Size, and Type

While a monster's size and type remain constant (unless changed by the application of templates or other unusual modifiers), alignment is far more fluid. The alignments listed for each monster represent the norm for those monsters—they can vary as you require them to in order to serve the needs of your campaign. Only in special cases is this more fixed, such as creatures with an Intelligence of 2 or lower (which are almost always neutral) and planar monsters (outsiders with alignments other than those listed are unusual and typically outcasts from their kind). Note "type" and "subtypes" are not capitalized. Each type and subtype should be appropriately linked, as many of them provide special abilities.

(emphasis mine)

The description here makes clear that while aligned outsiders are fairly fixed in their native alignment, they can change. Fallen angels, repentant fiends, rogue robots, and so on.

Dungeons and Dragons certainly made occasional plot points of such anomalies, such as Falls-from-Grace from Planescape: Torment (LN succubus), and even a demonic paladin. As discussed in a similar question for 3.5, the rules are a little more explicit about how this works, however.