1) By RAW, the player says what their character attempts to do (PHB p.6). By RAW, I can't find anything to suggest that the DM can control a PC's actions (while certain effects may allow an NPC to control another character's actions, that's not quite the same). So your druid could certainly attempt to wear metal armor.
2) If there is a chance of failure, the DM calls for an ability check (PHB p.174; see also, @iserith's Ph.D. thesis, Adjudicating Actions). It's hard to see how the druid would face any physical chance of failure putting on armor. But some DMs might rule that when a character attempts to do something against their alignment, bonds, ideals, etc., that the character needs to make a Wisdom check in order to force themselves to do it. However, I would argue that this violates point #1, above: the failure condition suggests that the DM, not the player, is saying what the character will attempt. That is, if a die roll represents a PC struggling against themselves, then they are both attempting to wear the armor and attempting NOT to wear the armor, and that second attempted action came from the DM, not the player. One could argue that the player did, in fact, announce an attempt to not wear metal armor when they chose the druid class, but that denies a player's right to change their mind, and at some point that line of reasoning will devolve into pure silliness.
3) Nor can I find anything to suggest that the DM fiat can spontaneously cause class features to stop working, which is another common response to a druid's attempt to wear metal armor. Obviously a DM can override RAW; but for that statement to make sense, it implies that there are things the DM can't do according to RAW (see D. Vincent Baker's Ph.D. thesis, Apocalypse World).
4) You know what the DM does control? THE ENTIRE REST OF THE MULTIVERSE (DMG p.1-320). So if a DM has an issue with a druid player wearing metal armor, they could control the reaction of NPCs, creatures, and even the natural world. Maybe animals shun the druid, and plants begin to wither in their presence. Maybe other druids are hostile and attack the apostate. Eventually, powerful fey or elementals or primordials might show up to eliminate the corrupt metal-wearing druid. Some DMs might use these techniques in a heavy-handed way to enforce their view on their players. But other DMs might mine it for rich story-telling potential. That's a matter of DM personality and is beyond the reach of RAW.
Your druid PC wearing metal armor? Then obviously they are not a druid. How do I know? Because the rules state clearly that druids won't wear metal armor. Period, end of story. If the PC can do anything the player says they can do, then my first level fighter can cast wish. While the DM doesn't make the rules for the PCs, the PHB and associated books do and they are clear. Want to run it differently in your home campaign? Feel free.
The DM enforces the rules, the players don't.