No, you should not pour grease down the toilet. The toilet and the kitchen sink drain into the same building sewer line, so grease poured into either one cools, hardens and coats the same pipe. Pour cooled grease into a can, let it set, and put it in the trash.
The toilet only looks like a shortcut. Its drain is wider, but it joins the same line under the house, and a flush is unheated tap water that cools grease fast. The North Texas Municipal Water District says fats, oils and grease “still eventually cool and harden wherever they are in your household plumbing or in wastewater system pipes,” and asks residents to flush only pee, poop and paper.
Grease does not stay alone in the pipe. It coats the wall and catches toilet paper, hair and wipes that never break down, and the mass keeps growing. Utilities call the large ones fatbergs, and the district warns they can back wastewater into homes or overflow into streets and waterways. A small amount each time still adds up, because every pour leaves another layer.
Instead, let drippings cool, pour them into a can, and trash it once solid. Wipe greasy pans with a paper towel before washing. Used frying oil can go to a city drop-off; the guide on getting rid of cooking grease lists Dallas and North Texas options. For food scraps, see what the disposal cannot take on Thanksgiving.
One honest limit: a toilet that flushes fine after grease went down does not mean the line is clear. The grease settles past the toilet, farther along the drain or the sewer line, and the buildup shows up later as slow drains, gurgling in other fixtures, or a backup.