To what does the phrase "the temple of the tabernacle of testimony" refer in Rev. 15:5?
The term "tent of witness"/"tabernacle of testimony" first appears in Numbers as a reference to the whole of the portable temple structure that the Levites were in charge of dismantling, transporting, re-erecting and camping round about it:
ESV Num 1:47 But the Levites were not listed along with them by their ancestral tribe. Num 1:48 For the LORD spoke to Moses, saying, Num 1:49 "Only the tribe of Levi you shall not list, and you shall not take a census of them among the people of Israel. Num 1:50 But appoint the Levites over the tabernacle of the testimony, and over all its furnishings, and over all that belongs to it. They are to carry the tabernacle and all its furnishings, and they shall take care of it and shall camp around the tabernacle. Num 1:51 When the tabernacle is to set out, the Levites shall take it down, and when the tabernacle is to be pitched, the Levites shall set it up. And if any outsider comes near, he shall be put to death. Num 1:52 The people of Israel shall pitch their tents by their companies, each man in his own camp and each man by his own standard. Num 1:53 But the Levites shall camp around the tabernacle of the testimony, so that there may be no wrath on the congregation of the people of Israel. And the Levites shall keep guard over the tabernacle of the testimony."
Within the borders of this traveling tent there was an area curtained off that was referred to as "the holy place". The "holy place" is what the original post is trying to identify. It is the "temple of" the tabernacle. This holy place was to contain the "covenant box" (aka "ark of the covenant") which contained the ratified Sinai covenant which served as a constant reminder that the Israelites were bound to the terms of the Suzerein treaty with Yehovah:
Deu_31:26 Take this book of the law, and put it in the side of the ark of the covenant of the LORD your God, that it may be there for a witness against thee.
Deu 31:24 When Moses had finished writing the words of this law in a book to the very end, Deu 31:25 Moses commanded the Levites who carried the ark of the covenant of the LORD, Deu 31:26 "Take this Book of the Law and put it by the side of the ark of the covenant of the LORD your God, that it may be there for a witness against you.
By having the text of the Torah travel with the Israelites where ever they went Moses would be assured that the Torah would constantly be rebuking them for their violation of the Torah and making certain that they know that God blames them for any evil he brings upon them in judgement!:
Deu 31:27 For I know how rebellious and stubborn you are. Behold, even today while I am yet alive with you, you have been rebellious against the LORD. How much more after my death! Deu 31:28 Assemble to me all the elders of your tribes and your officers, that I may speak these words in their ears and call heaven and earth to witness against them. Deu 31:29 For I know that after my death you will surely act corruptly and turn aside from the way that I have commanded you. And in the days to come evil will befall you, because you will do what is evil in the sight of the LORD, provoking him to anger through the work of your hands."
The "covenant box" is also referred to as the "box of testimony":
Exo 25:10 "They shall make an ark [box] of acacia wood. Two cubits and a half shall be its length, a cubit and a half its breadth, and a cubit and a half its height. Exo 25:11 You shall overlay it with pure gold, inside and outside shall you overlay it, and you shall make on it a molding of gold around it. Exo 25:12 You shall cast four rings of gold for it and put them on its four feet, two rings on the one side of it, and two rings on the other side of it. Exo 25:13 You shall make poles of acacia wood and overlay them with gold. Exo 25:14 And you shall put the poles into the rings on the sides of the ark to carry the ark by them. Exo 25:15 The poles shall remain in the rings of the ark; they shall not be taken from it. Exo 25:16 And you shall put into the ark the testimony that I shall give you.
The "testimony" placed in the box was the Torah, represented by the two broken flat rocks into which were inscribed the "10 sayings" along with a a couple of other tokens:
Heb 9:3 And after the second veil, the tabernacle which is called the Holiest of all; Heb 9:4 Which had the golden censer, and the ark of the covenant overlaid round about with gold, wherein was the golden pot that had manna, and Aaron's rod that budded, and the tables of the covenant;
But by the time Solomon's temple was built it only contained the two stone tables:
1Ki 8:9 There was nothing in the ark except the two tablets of stone that Moses put there at Horeb, where the LORD made a covenant with the people of Israel, when they came out of the land of Egypt.
The "holy place" of the temple structure was where the covenant box was placed:
Heb 9:1 Now even the first covenant had regulations for worship and an earthly place of holiness. Heb 9:2 For a tent was prepared, the first section, in which were the lampstand and the table and the bread of the Presence. It is called the Holy Place. Heb 9:3 Behind the second curtain was a second section called the Most Holy Place, Heb 9:4 having the golden altar of incense and the ark of the covenant covered on all sides with gold, in which was a golden urn holding the manna, and Aaron's staff that budded, and the tablets of the covenant. Heb 9:5 Above it were the cherubim of glory overshadowing the mercy seat. Of these things we cannot now speak in detail.
The design of the tabernacle was to be made to imitate the true temple in the sky:
Exo 25:8 And let them make me a sanctuary, that I may dwell in their midst. Exo 25:9 Exactly as I show you concerning the pattern of the tabernacle, and of all its furniture, so you shall make it.
Act_7:44 "Our fathers had the tent of witness in the wilderness, just as he who spoke to Moses directed him to make it, according to the pattern that he had seen.
Heb 9:23 It was therefore necessary that the patterns of things in the heavens should be purified with these; but the heavenly things themselves with better sacrifices than these. Heb 9:24 For Christ is not entered into the holy places made with hands, which are the figures of the true; but into heaven itself, now to appear in the presence of God for us:
The box of testimony was lost when Solomon's temple was destroyed but John sees the curtain to the holiest place pulled back and the true temple contains the true box of testimony:
Rev 11:19 Then God's temple in heaven was opened, and the ark of his covenant was seen within his temple. There were flashes of lightning, rumblings, peals of thunder, an earthquake, and heavy hail.
It may be that the box now contains the new covenant and its blessings rather than the curses of the Torah though that is not explicitly stated. It might be that the reference to scary "peals of thunder and an earthquake" etc. suggest that it still to be understood to contain the tablets.
Actually; Exo 35:10 - 38:31 describes exactly what it is and it's construction. Also in ch 38, there's some really interesting passages at the end of it's description. Specifically, "The Cost of the Tabernacle", is revelation to what you're asking about in Rev 15:5-8 in light to what's stated in those passages (and in earlier comments).