If we're talking directly about the positions vs. TULIP, then the only thing that is retained is Total Depravity - but in the same way Arminius himself accepted Depravity, with prevenient grace given to all to allow the choice to follow or deny. So to go along with that, the UMC rejects predestination (and those things which imply or lead to it like Limited Atonement) and once saved always saved. Views on how God's mercy is poured out in salvific issues beyond that is a mixed bag.
If you mean some of the secondary positions, like Postmillennialism or views on the nature of Communion or Baptism, then that's a more open-ended question. There is no set end times view, although you will probably notice that Methodists tend to reject Dispensationalism (the theology in "Left Behind", in other words). Communion and Baptism are held as sacraments in denoms like Presbyterianism, but not Baptist churches. Methodists view those as sacraments, outward signs of inward grace initiated by God, and not as ordinances observed by Man.
Of course, this probably doesn't help, but there are Reformed Methodists that do follow in the Calvinistic tradition. They're in a severe minority and isolated around Wales, but they do exist. And it goes without saying that they aren't affiliated with the UMC.