There are 2 things here.
I think going outside of range shouldn't break concentration, shouldn't end the spell, BUT if the effect requires an action/bonus action to activate it then it should require sight (and/or range).
I don't think Player's Handbook p.203 is enough here.
Once a spell is cast, its effects aren’t limited by its range, unless the spell’s description says otherwise.
It only talks about range, and spells in general. Grease, Web, Alarm... shouldn't require the caster to stay in range (or have Line of Sight), but they also don't require action to keep their effects going.
The important quote here is "unless the spell’s description says otherwise."
It doesn't say "unless the spell’s description specifically limit the effect's range.". "says otherwise" is pretty vague. Using an action to reactivate the effect can be interpreted as "otherwise", since the effect isn't applied automatically.
The Sage Advice Rules of Spellcasting article isn't specific enough either.
You don’t need to be within line of sight or within range to maintain concentration on a spell, unless a spell’s description or other game feature says otherwise.
It does specify that PHB 203 applies for concentration spells too, and nor range nor LoS is limiting concentration. (Grease and Alarm isn't concentration spells, but Web is.)
In contrast Arcane Eye specifically says that it can be moved at any range.
As an action, you can move the eye up to 30 feet in any direction. There is no limit to how far away from you the eye can move, but it can’t enter another plane of existence.
If the general PHB 203 rule would be enough to not limit range, then why does the spell specify that the action isn't limited by range?
Neither the Heat Metal spell description nor the quoted rules above specific enough to make a definite ruling IMHO. If it didn't require any action to keep the metal heated then subsequent damage wouldn't require range or LoS, but because it does I feel it should.
(If it required an action, then at least there would be a chance to catch up with the caster, but because it's a bonus action the caster can get out of range or out of sight from even ranged opponents as well. That's just too powerful for a 2nd level spell - for the mods against the players - when the 4th level Arcane Eye requires an actual action...)