If the Earth lost its tilt, would lands currently at the equator become cooler?

No, no, and no. There would be no seasons at all. All climates would stabilize to an average and remain there, hottest at the equator, coldest at the poles, the Earth itself would not be any closer to the sun. The climate at the equator would get slightly warmer overall, and stay that way all year. The climate at the tropics would be an average between its current summer and winter temperatures. The climate at both poles would be mostly colder, but perhaps not quite as cold as it is now in deep winter. Also you would no longer have several months straight of darkness and light at the poles. If you were standing exactly at the north or south pole, the Sun would look like it was skimming the horizon 24 hours a day, all year long. Like permanent twilight.