I looked at the Ticketmaster site, and they list no more tickets after about the 3rd week in June. However, another site I consulted wrote that the play's end was open-ended. Anyone know for sure?

No official closing date has been posted. Even the fact that Ticketmaster is selling tickets through the 3rd week of June does not guarantee that the show won't close before then. The show has been reporting as low as 50% in attendance. That's usually a sure bet it's going to be closing fairly soon.
Why do you ask? Are you looking for tickets beyond that date?

Yes, I'm looking for tickets for July, the first week of...

I'd think if they are extending it, they'll probably do so by about the first of February.

When a long-running is not close to selling out, it's not going to be selling tickets more than, say, five or six months in advance for an obvious reason: It may not last that long. At the same time, however, it doesn't mean that the show will close soon by then. It may...or it may not!
In the case of Rent, except for a spurt when the movie version came out last year, the show has been running far below capacity for quite a while. So, again, the answer is: It may be still be playing in July...or it may not be playing in July!
Either way, you'll just have to wait a while. Meanwhile, don't fret. If it's still playing, you'll have no problem getting tickets....and probably at a discount!

Don't know if the show will still be around when July comes, but you can assume two things:
1. It's more than likely that the show will not become a hit till July, (if it's still showing) so discount tickets will be available.

2. That written, keep tabs on Broadwaybox.com during the months untill July comes - If the show is still running, they'll probably have discount tickets for it.

Only a soothsayer would know. Producers can close non-profitable shows on short notice. I agree with anybody who told you to wait to buy tickets. If the show is still running in June, you may be able to get half-price seats at the TKS Office in Times Square.