My last stop on an excursion to Jordan is the Jordan Valley Marriott Hotel, on Dead Sea Road, on the Jordan side of the Dead Sea in Sweimeh, Jordan (well south of Amman). I need to get from that hotel back to the airport at Tel Aviv, Israel (TLV). Is there any bus or other public transport to do that?
Perhaps the Marriott would be the best source for this information.
If you don't get the answer you need here, look on the Israel Forum of TripAdvisor: https://www.tripadvisor.com/ShowForum-g293977-i1733-Israel.html
If you're looking on a computer and not a phone, there are FAQ's on the right side of the page I linked; several of them are about border crossings.
It seems that getting between Jordan and Israel takes time, and the exact amount of time cannot be readily predicted, so you may have to do it the day before your flight to ensure you don't miss it.
Okay, you're at the north end of the Dead Sea. The land option is local transport (taxi probably) to the Allenby Bridge border crossing. Presumably you are traveling from Israel to Jordan so you will have an Israeli visitor's visa already. They are issued on a small piece of sturdy paper rather than stamped in your passport. Don't lose it! I am not sure if you can get a tourist visa at that border crossing - it's used primarily by locals and diplomats, not much by tourists. I believe there are buses that cross the border and continue to Jerusalem (Damascus Gate bus station). There are certainly taxis and there may be buses from the Israeli side as well. The border crossing can take a few minutes or several hours - there's no pattern to it, though I would expect delays to be less likely around midday. Once at Damascus Gate, you'll have to get through part of traffic-congested Jerusalem to the central bus station for a bus to TLV. That ride takes a little over an hour BUT the bus does not enter the airport. You'd have to walk some distance (less than I km, I think) to the old terminal (T1) and then take a shuttle to T3. Or you could order a Nesher shared taxi (at least 24 hours in advance) to pick you up at some place in Jerusalem (Damascus Gate bus station is not a good option, maybe one of the hotels or restaurants nearby. ) The shared taxi makes several stops to pick up passengers, so you may have a longish journey if you happen to be first or second to be picked up, which is impossible to know until you're in the taxi. It's about a 45 minute drive to TLV once all the passengers are aboard and it will go directly to T3. For more information about transport in Jordan, go to the Jordanian forum at tripadvisor.com. Note that the border crossing closes at 3 p.m. on Fridays and Saturdays. In looking for something else, I came across this service for transport from Jordan to Jerusalem that may be useful to you and has a description of what to expect.
As you may have figured out, that's a lousy option. The only other one is to get to Amman airport and take a Royal Jordanian flight to TLV. If you plan to connect at the airport, allow plenty of time since you'll have to go through passport controls, collect your luggage and then check in for your next flight. The only ray of sunshine is that you'll arrive and depart from T3. It is strongly recommended to allow at least 3 hours for check-in - that's 3 hours after you have gone through passport control and collected your luggage.
If you fly from Amman and need to stay overnight in Israel before your flight from TLV, your options are Tel Aviv or Jerusalem. It's definitely faster to get to/from TLV from Tel Aviv. There are trains at least once an hour between TLV and the city throughout the night. The ride is fast and cheap. Then it's a short taxi or bus ride, or longish walk to a hotel for the night. The trains stop running a couple hours before sunset on Friday and resume service on a couple hours after sunset on Saturday night. Taxis cost more - there's a fixed fare. Jerusalem is somewhat farther. A shared Nesher taxi will take longer than a "regular" taxi, but will cost less than the Tel Aviv taxi. You can get more info on this using the forum Harold's given you a link to. If you choose the land option, you are better off staying the night in Jerusalem.