Maybe, as long as it keeps talking non-stop forever
How does Voice of the Chain Master work?
The Voice of the Chain Master eldritch invocation reads:
You can communicate telepathically with your familiar and perceive through your familiar’s senses as long as you are on the same plane of existence. Additionally, while perceiving through your familiar’s senses, you can also speak through your familiar in your own voice, even if your familiar is normally incapable of speech.
There are several basic things to note here:
You speak through the familiar. It's not your familiar speaking on your behalf or anything like that, it's you actually talking with your own voice, just in a remote location and "through" your familiar.
You can do this 'while perceiving through your familiar's senses'. Ordinarily with Voice of the Chain Master, that's limited to being on the same plane of existence, but that's not actually a requirement if you have some other way to perceive through your familiar's senses while it's off-plane.
You aren't (necessarily) where your familiar is, you are just speaking from that location without actually being there
How does Diplomat work?
The third benefit of the Diplomat feat reads:
If you spend 1 minute talking to someone who can understand what you say, you can make a Charisma (Persuasion) check contested by the creature's Wisdom (Insight) check. If you or your companions are fighting the creature, your check automatically fails. If your check succeeds, the target is charmed by you as long as it remains within 60 feet of you and for 1 minute thereafter.
There are some basic things to note here as well:
Diplomat expects but does not require a target within 60 feet of you at the end of the 1-minute timer.
Diplomat works weird linguistically on a target who was never within 60' of you.
You can use the feat any time you spend a minute talking to someone who can understand you, regardless of distance or attitude or whatever else. You don't need to be present, though c.f. If you're listening to a voice recording, can you hear me? .
How does Diplomat work against targets beyond 60'?
The language here is ambiguous. 'As long as' is clearly being used in the temporal rather than conditional sense, given the following 'and for 1 minute thereafter', but 'thereafter' doesn't have a clear meaning in the case of a target that isn't within 60 feet. There's no consistent standard for how to read the sentence in English in that case, but the two most common readings of similar sentences in my experience are that either the thing that didn't happen is treated as if it happened for 0 seconds (i.e. the 'additional time' is the amount of time expected to be used) or the total time is now expected to be 0 (i.e. the 'additional time' is lost).
In this case, the best for the warlock is that it lasts the 1 extra minute when having a conversation at 65' distance. That's also what happens if you interpret "thereafter" to mean after the base time of 0 rather than the remaining specifically.
Putting it all together
Voice of the Chain Master definitely lets you use the Diplomat feat at any range, bypassing the Treachery of Images issue with recordings. However, the ability when used beyond 60 ft certainly only last for a minute, and may if the DM is feeling mean not last more than 0 time-- the charmed condition being applied and removed all at once, useful only for things that trigger off of a creature becoming charmed by you.
Since the effect only lasts a minute and takes a minute to set up, keeping it up continually would require a constant stream of talking from your familiar-- consuming your senses and action each round, and likely being really obvious, suspicious, and annoying.
Note also that while your victim can't attack you while charmed, they can totally attack your familiar.