My HP 1505 is saying it has a paper jam, but there is no paper jam. I have turned it off and it still says it has a paper jam.

If you check closely, you may find there is a small piece of paper trapped between the rollers?
Good Luck!
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There is definitely a paper jam. It's up to you to find it. Sometimes it's a very small piece of paper. I have worked with these printers before. If it says paper jam, that means there is a paper jam. Look a little bit more closely. I garranty you 100% there is a paperjam.

check paper sensor, it will find back of the printer.

Below is the exact link from HP to resolve Paper jam, but if you have tired everything, the printer may need replacement: Paper Jams or Paper Feed Problems

off your printer and manually feed paper by turning roller as guide, observe if sensor key freely move when paper pass, some sensor key stock when forcefully remove the jam paper inside.

Paper jams can be tricky to solve. The paper is usually stuck inside the printer mechanism so even if there is no visible paper jam, you will still get the paper jam error. Try this:

1. Turn off your printer
2. Open all trays and remove any paper stuck there
3. Locate the button to open the printer, open it and remove any paper stuck there also.
4. Repeate the above step for all buttons that you can find to access the inside of the printer
5. Close all doors and put back all trays
6. Turn on your printer

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maybe there is more than 1 piece of paper jamed

Try unplugging the printer from power, (Turned off of course) Wait 5 minutes, plug the printer back in. Now open the case like you're going to clear a paper jam, or change ink cartridges. Close the case. Press the Power On button of the printer.

Do a complete reset.
1- Clear the printer queue.
2- Turn off the device.
3- Remove paper in the tray and possible paper jammed.
4- Remove ink toner.
5- Turn on the device briefly and turn it off.
6- Replace paper and toner.
7- Turn on!

check the paper sensor maybe one of the sensor stuck