With the DFA process, the player is immediately removed from the 40 man roster. The team has something like 10 days to put the guy through waivers or to deal him. After that, the player can be outrighted to the minors. If the player has been in the majors in enough seasons, then he can refuse assignment to the minors and become a free agent. My guess is that Weems cannot refuse the outright, without checking his past service (dunno how many times he was up and down before the nats picked him up). Ramirez, I'm guessing, might actually be tradeable since he's hit lefties well. IF not, then I think he may have enough time in to become a free agent.
Weems had 2.055 at the opening of this season and has been on the MLB roster the whole season. That should get him well past 2.172 = 3, so he should be able to reject.
Ramirez had 4.124 at the start of the season, so he's well past being outrighted without his consent. But I don't see the path to trading him. He'd be arb eligible if he were on the 40, but without being on the 40 he could just become a minor league free agent. I don't see why this DFA wouldn't result in him choosing free agency now.
He can either (1) go to Rochester and wait for an opportunity to get back to DC somehow, knowing that he's behind anyone they want for a September callup from the 40 man; (2) elect FA, sit on his ass and collect his entire salary; (3) elect FA and possibly sign with someone else, for whom he'd be eligible for the playoff roster and then salary arbitration in the winter, although realistically he'd probably be non-tendered. One of those options seems much worse than the others.