"I'll show you to your room" means I'll take you to your room.
The question is: Can "to" be replaced by "into"?
I have done a quick search and found out that "show someone into" is uncommon and it wasn't clear whether they are equal.
My guess though is that "into" takes someone inside the room. But if so still what is the difference? Is there something special about entering the room with them?
Edit:
Macmillan's entry for show, gives this sense, 5:
[transitive] to lead someone somewhere, for example because they do not know where to go
show someone to something:
Let me show you to your room.
show someone into something:
She showed me into a sunny room where two children were playing.
A quick search on Google books, gives me this sentence from "Jack Archer" By G. A. Henty:
Two servants will show you into your carriage.
I can get by the set phrase but it is the curiosity while I am at it.
Best Answer
Native speakers simply wouldn't normally say "I'll show you into to your room". The two credible possibilities are...
In practice, both contexts often apply simultaneously, and either of the above phrasings could be both intended and understood as implying the other.
For reasons that aren't immediately obvious to me, although I've said into is unlikely in OP's exact context (with a room), it seems perfectly normal to me to...
EDIT: In informal contexts such as a guest staying overnight at your house, #2 above (no preposition) is the normal form. In formal contexts (hotel staff, a wealthy person's house-servants, etc.) to is more common, carrying either/both implications of guiding and/or escorting.
The relative uncommon into (almost always formal) implies the speaker will accompany you through the entrance (door, gate, etc.) to wherever you're going, and assist you in "settling in". Thus, anyone showing you into the garden would usually accompany you into the garden and introduce you to your host and/or other guests. If a hotel manager shows you to the dining room, he might well just leave you at the doorway - but if he shows you into the room, he'll probably usher you to a table and see that you're seated before leaving you in the care of the restaurant staff.