You need to mount the directory above the installation directory. I.e.:
mount c C:\DOSGames\DOSGAMES
c:
cd tiecd
tie

Tie Fighter installer (at least Collector's CD-ROM version) writes C:TIE.CD file that tells where it is installed. In your case and probably by default it would contain C:\TIECD

Moved to the Dosbox forums, next time you have a problem, please post there 😉

Still getting the installation screen after typing
mount c C:\DOSGames\DOSGAMES
c:
cd tiecd
tie

It is the Collector's CD-ROM.
I'm going to try a reinstallation and see if I can change that "C:\TIECD" command.

Nope!
It didn't even let me reinstall it.
I had to retrieve the game files out of the recycle bin and put them back in the "DOSGAMES" folder.
I'm still getting the same installation screen instead of gameplay.

Also, I found a "TIE.CD" file outside the "DOSGAMES" folder.
Should I move it?

I don't know how your Tie Fighter files could possibly have ended up in Windows' Recyling Bin, but how about you start over and explain step by step how you're installing the game including all the mount commands.

The game has installed and worked perfectly fine for me this way:
- mount some existing directory as C,
- mount the optical drive, Daemon Tools virtual drive, or image file as D
- in Dosbox, switch to D: and run the installer
- after installing, switch to C:, cd into the game directory (C:\TIECD by default if I remember correctly) and run the game
- in the future, always use the exact same mounts as when the game was installed

edit: you can even run the installer from the CD and it will give you the option to play the game, no need to CD to C:\TIECD.

The commands I used to install the game:

Mount d D:\ -t cdrom
d:
install

Installed using "maximum install" and "express install".
Music and sound testing worked.
I clicked "exit and run Tie Fighter".
Game worked, but when I quit and tried to play it again, the "install" screen came up.

Using the commands, "mount d D:\ -cdrom", "mount c C:\DOSGames\DOSGAMES", "D:", "install" I got that screen your showing me, but when I clicked "play", I got this error message:
ERROR #3: You must insert Tie Fighter CD into your CD ROM drive. If the CD is already in the drive and you still get this error, then check to make sure that your CD ROM drivers are loaded.

When I clicked "install" on that same screen, all the "install" options were disabled whether the game was installed or not.

I understand "switch to c:", but I don't understand "cd into the game directory (C:\TIECD by default if I remember correctly)".

The commands I used to install the game: […]
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The commands I used to install the game:

Mount d D:\ -t cdrom
d:
install

So you did not mount C: at all? Then the game did not get installed. You have to mount C: also when you install the game, not just when playing afterwards.

Using the commands, "mount d D:\ -cdrom", "mount c C:\DOSGames\DOSGAMES", "D:", "install" I got that screen your showing me, but when I clicked "play", I got this error message:
ERROR #3: You must insert Tie Fighter CD into your CD ROM drive. If the CD is already in the drive and you still get this error, then check to make sure that your CD ROM drivers are loaded.

Your D mount command is missing the '-t' before 'cdrom'.

I understand "switch to c:", but I don't understand "cd into the game directory (C:\TIECD by default if I remember correctly)".]

c: <--- switches to c:
cd tiecd <--- switches to c:\tiecd
But you don't need to do this since the installer also allows you to play.

IT WORKS!!!!!
Thank you! And thank the good Lord in Heaven!

Did a fresh reinstallation with "c C:\DOSGames\DOSGAMES" mounted.
And the "cd tiecd" command works now.
After that, I hit "Tie", and the game begins.

Again, thanks a bunch!

Enjoy the best game ever released 😀