Let's work through these options systematically.
First, let's just throw out the idea of Ramos moving to third base. That's not going to happen. Because it can be discussed as a realistic option for Mauer, many people seem to have the perception that any catcher can easily be shifted to third base. That's not the case. With his big frame and strong arm, Ramos is perfectly built to catch, but he's slower than molasses, lacking the quickness and athleticism needed to man the hot corner. Throughout his minor-league career, Ramos has played catcher almost exclusively despite the Twins having Mauer entrenched. There's a reason for that: Catcher is Ramos' natural position and it's not going to change.
Aside from his not being athletically equipped to play the position defensively, a move to third would drastically lower Ramos' overall value. Even if he could field the position adequately, he'd almost certainly be far worse there than he is at his natural position of catcher, changing his defense from a strength to a liability. His bat also wouldn't play very well at third, because while some folks are mesmerized by his hot start with the Twins, it can't be forgotten that he's registered a relatively modest .288/.336/.441 hitting line with only 34 home runs in 302 minor-league games. Now, he's been young for every level coming up and it's hardly a stretch to think he's capable of improving on that line as a big-leaguer -- particularly after seeing him launch balls all over the place in his first few games -- but Ramos doesn't possess an elite bat that absolutely must be in the lineup. He's a great-hitting catcher, falling in the Bengie Molina mold.
It's for that same reason that giving Ramos significant time at designated hitter, as Seth suggested, is not a legitimate option. Nor is shifting him to first base, which is the only other defensive position he'd likely be able to handle well. Ramos hits very well for a catcher but there's no reason to think he's a strong enough hitter to be an asset at first base or DH.
The only scenario in which keeping Ramos long-term makes sense is if Mauer is moved to third base. Unlike Ramos, Mauer possesses the athleticism and offensive aptitude to make such a move palatable. But that's a played-out debate; Mauer is an elite player and much of his value is derived from his ability to play the catcher position at a Gold Glove caliber level. The Twins shouldn't even be thinking about moving Mauer out from behind the plate for several years.
Ramos is valuable because he's a great young defensive catcher with pop. Those are hard to find around the league. But unfortunately the Twins are set at catcher for the foreseeable future and -- as I've outlined above -- the organization will be lessening the value of a great asset if they try to fool around with switching Ramos' position.
While keeping Ramos around as a backup who can fill in for Mauer and DH against lefties occasionally would be a nice luxury, this team has other legitimate holes that need to be filled going forward and Ramos is their best chit to accomplish that. The dearth of great catchers around the league makes Ramos a very enticing piece, and if he's smart Bill Smith will start gauging interest as the deadline nears from teams that have organizational strengths at second base, third base and starting pitcher.
It's rare that a prospect as promising as Ramos is expendable, but for the Twins, he is. He has the look of a guy who should be catching full-time in this league and the Twins already have their catcher entrenched until Ramos is past 30.
Simply put, Ramos is more valuable to most other teams around the league than he is to the Twins. For that reason, trading him -- whether at the deadline this year or after the season -- is essentially a no-brainer.