Are the lights in the unit still working? If not it is a power issue. A lot of the vintage gear has a fuse inside that is not user replaceable. It could be the sensing circuit or an overload relay failure too. Unless you know electronic repair, have parts suppliers available and the equipment to service it, you are going to have to send it in
You will need an engineer for this. The volume control is on these machines controlled by an IC or chip. That sometimes goes faulty and needs to be replaced. They have often a great deal of pins that solder on to the board and need special tools to get them on and off.
If you can contact Denon to see if they have a repair centre in your part of the world.
As going inside the unit would end any guarantee. If you cannot get satisfaction from them and are prepared to violate the guarantee terms, then open up and look to the power supply section. With a bit of luck the fuse might have failed. But you will have to foot the bill for all future repairs.
Try pressing the Band button on the remote or on the AVR-1604. On the receiver, the band button is just below the left-hand edge of the LCD screen. To the right of the band button are the DTS button and the preset station buttons. There are no warnings about needing both the FM and AM antennas connected in the manual. Crutchfield has the manual for the AVR-1604 on the product page: https://www.crutchfield.com/S-uT0aFEwnSpF/p_033AVR1604/Denon-AVR-1604.html . The current link to the manual is https://pdf.crutchfieldonline.com/ImageBank/v20111011085300/Manuals/033/033AVR1604.PDF .
Denon's support site, https://denon.custhelp.com/app/answers/detail/a_id/3954/~/am%2Ffm-%28tuner%29-audio-reception , recommends having both antennas connected. A replacement FM antenna is fairly cheap. This one, https://www.amazon.com/Ancable-Replacement-Antenna-AVR-3806-3808CI/dp/B01LW5A7AU , looks similar to the one included with the original parts for the AVR-1604.
I hope this helps.
