If the Menorah or any other Temple Articles were theoretically found or any of the other less significant ones like the tongs used to change the wicks on the daily basis, would it be allowed to be displayed in a museum-type place or would that be against Halacha?
You cannot benefit (hana'a) from items of Hekdesh or you may violate the Biblical prohibition of meilah. Viewing an item would not be meilah, but may be forbidden rabbinically if it is avoidable.
I'm not sure if viewing an ancient item in a museum would be considered hana'a, but if it is, then it seems it would be rabbinically forbidden.
This means even viewing the Temple could be problematic if its done for your own hana'a and is avoidable! So displaying the items in a museum may be forbidden too.
(According to the first version of the gemara in Pesachim, there is no prohibition on viewing hekdesh, but the halacha follows the second version.)
Note that the stones to the walls of Har Habayis (such as the Western Wall) may be Hekdesh also. The Romans knocked down many of them and they remain on the ground today. So don't use them to build a house or even sit on!
Update: Answer changed based on Seth J's source.