I'm trying to install Linux (any linux, but preferrably Ubuntu Mate) onto my cheap tablet Teclast X80H, which came preinstalled with WIN8/Android x86.
Main specs are:
CPU: Intel Bay Trail, 1830 MHz, Cores: 4
GPU: Intel HD Graphics, 646 MHz
RAM: 2 GB, 1333 MHz
Storage: 32 GB
Display: 8 in, IPS, 1280 x 800 pixels, 24 bit
Tablet is locked on 32 bit UEFI, so I worked with 64 bit distributions adding bootia32.efi, which allows most distributions to boot.
I tried many latest flavours of linux - Ubuntu 15.10, Ubuntu Mate 15.10, Slackware 14.1, Opensuse 20151111, Linux Mint 17.2, Linux mint 17.3, Linux Mint debian edition 201503, Fedora 23.10.
Most I was able to boot into grub and tried running live version to test. Main problem seem to be, that latest distributions don't recognize tablet's display.
I will continue description on Ubuntu 15.10, but I suspect others behave the same.
In Ubuntu I discovered (having previously read on some forums similar problems), that even that there is no picture on tablet's display (after selecting live on grub), HDMI is active and so live OS comes succesfully onto desktop, which works quite well (also touchscreen and wifi work).
Still no picture on tablet's display.

xrandr --query outputs following:
Screen 0: minimum 8 x 8, current 2720 x 1280, maximum 32767 x 32767
DSI1 connected primary 800x1280+0+0 (normal left inverted right x axis y axis) 0mm x 0mm
800x1280 60.00*+
DP1 disconnected (normal left inverted right x axis y axis)
DP2 disconnected (normal left inverted right x axis y axis)
HDMI1 connected 1920x1080+800+0 (normal left inverted right x axis y axis) 531mm x 299mm
1920x1080 60.00*+ 50.00 59.94
1920x1080i 60.00 50.00 59.94
1600x1200 60.00
1680x1050 59.88
1280x1024 75.02 60.02
1440x900 74.98 59.90
1280x960 60.00
1280x800 59.91
1152x864 75.00
1280x720 60.00 50.00 59.94
1024x768 75.08 70.07 60.00
832x624 74.55
800x600 72.19 75.00 60.32 56.25
720x576 50.00
720x480 60.00 59.94
640x480 75.00 72.81 66.67 60.00 59.94
720x400 70.08
HDMI2 disconnected (normal left inverted right x axis y axis)
VIRTUAL1 disconnected (normal left inverted right x axis y axis)

I tried playing with different kernel boot options in grub, without success. No picture on tablet's display.

Interesting info. Running live version of Linux Mint debian edition display's picture on tablet's display nicely. Unfortunately due to ancient kernel, no touchscreen and no wifi (probable other peripherals as well).

xrandr --query here outputs following:

Screen 0: minimum 320 x 200, current 800 x 1280, maximum 8192 x 8192
HDMI1 disconnected (normal left inverted right x axis y axis)
DP1 disconnected (normal left inverted right x axis y axis)
HDMI2 disconnected (normal left inverted right x axis y axis)
DP2 disconnected (normal left inverted right x axis y axis)
UNKNOWN1 connected 800x1280+0+0 (normal left inverted right x axis y axis) 0mm x 0mm
800x1280 60.00*+

I'd appreciate any help.

have you tried manually setting the resolution using xrandr? i assume you're getting this output from one of the TTY's?

Anyway, I described on this link, how I managed to install usable Debian onto my device:
http://pastebin.com/TBtrpE8R