Most common cause of this is if your display is connected via an analogue connection. Because it changes screen resolution the display has to re-adjust for the change in resolution. It some times fails to do this automatically.
To make it clear VGA connectors are the most common analogue video source for computer monitors. Other analogue connections include composite, component, SCART, etc. If you are using DVI or HDMI they are usually (but not always) digital. Digital connections do not suffer adjustment problems because the data they transmit is better defined (fewer ambiguities).
You should also run Diablo II in compatibility modes. In Windows 7 these included "Disable visual themes", "Disable desktop composition" and Windows XP Service Pack 3 compatibility mode. Not all of these may be relevant to Windows 10.
You can also try a third-party Voodoo GPU emulator driver. Many people swear these give the best graphics and performance for Diablo II.
Do note that Direct3D mode of Diablo II can bug on some modern GPUs. Not only can the graphics appear messed up but they can also perform very badly. For example you might get frame rate issues where the game drops to <5 FPS. This is on modern computer systems capable of running games like StarCraft II and Diablo III at consistently >30 FPS. As such it is recommended to use DirectDraw display mode for Diablo II if you do not want to look into Voodoo GPU emulators.
EDIT:
I can confirm Diablo II works flawlessly on Windows 10.