Does the Bible indicate whether Solomon is in heaven?

The Bible has been interpreted in different ways on this issue. A Calvinist perspective may be particularly inclined to take passages such as 2 Samuel 7:11-16:

...“‘The Lord declares to you that the Lord himself will establish a house for you: 12 When your days are over and you rest with your ancestors, I will raise up your offspring to succeed you, your own flesh and blood, and I will establish his kingdom. 13 He is the one who will build a house for my Name, and I will establish the throne of his kingdom forever. 14 I will be his father, and he will be my son. When he does wrong, I will punish him with a rod wielded by men, with floggings inflicted by human hands. 15 But my love will never be taken away from him, as I took it away from Saul, whom I removed from before you. 16 Your house and your kingdom will endure forever before me; your throne will be established forever.’”

(quoted in Benjamin Hoogterp's answer) and 1 Kings 3:3a:

Solomon showed his love for the Lord by walking according to the instructions given him by his father David...

together with their doctrine of Perseverence of the Saints to definitively settle the matter in the affirmative.

However, as per Mojo's comment, the promise in 2 Samuel is perhaps best interpreted as applicable to rule over the Kingdom of Israel and not Solomon's personal salvation; particularly in the light of other passages such as:

9 “And you, my son Solomon, acknowledge the God of your father, and serve him with wholehearted devotion and with a willing mind, for the Lord searches every heart and understands every desire and every thought. If you seek him, he will be found by you; but if you forsake him, he will reject you forever. - 1 Chronicles 28:9 NIV (emphasis added)

...except that he offered sacrifices and burned incense on the high places. - 1 Kings 3:3b (emphasis added)

King Solomon, however, loved many foreign women besides Pharaoh’s daughter—Moabites, Ammonites, Edomites, Sidonians and Hittites. 2 They were from nations about which the Lord had told the Israelites, “You must not intermarry with them, because they will surely turn your hearts after their gods.” Nevertheless, Solomon held fast to them in love. 3 He had seven hundred wives of royal birth and three hundred concubines, and his wives led him astray. 4 As Solomon grew old, his wives turned his heart after other gods, and his heart was not fully devoted to the Lord his God, as the heart of David his father had been. 5 He followed Ashtoreth the goddess of the Sidonians, and Molek the detestable god of the Ammonites. 6 So Solomon did evil in the eyes of the Lord; he did not follow the Lord completely, as David his father had done.

7 On a hill east of Jerusalem, Solomon built a high place for Chemosh the detestable god of Moab, and for Molek the detestable god of the Ammonites. 8 He did the same for all his foreign wives, who burned incense and offered sacrifices to their gods.

9 The Lord became angry with Solomon because his heart had turned away from the Lord, the God of Israel, who had appeared to him twice. 10 Although he had forbidden Solomon to follow other gods, Solomon did not keep the Lord’s command. 11 So the Lord said to Solomon, “Since this is your attitude and you have not kept my covenant and my decrees, which I commanded you, I will most certainly tear the kingdom away from you and give it to one of your subordinates. 12 Nevertheless, for the sake of David your father, I will not do it during your lifetime. I will tear it out of the hand of your son. 13 Yet I will not tear the whole kingdom from him, but will give him one tribe for the sake of David my servant and for the sake of Jerusalem, which I have chosen.” - 1 Kings 11:1-13 NIV (emphasis added)

If David's warning was inspired by God, and the description of Solomon's heart turning away from the Lord actually denotes a 'forsaking' of the Lord (which interpretation is entirely supported by the prophetic word of Ahijah to Jeroboam - cf. 1 Kings 11:33 in context), and unlike Manasseh (whose late-in-life repentance is documented) he did not repent, then an entirely reasonable (and my own Wesleyan-Arminian) interpretation is that, unfortunately Solomon is actually an example of an apostate and has not gone on to a heavenly reward - this seems to be the perspective of Wesley himself:

The Scriptures seem to name many cases of those who once knew God and fatally fell...As David declared, Solomon forsook God and was "cast off forever" though in his early life "Solomon loved Jehovah, walking in the statutes of David his father." 1 Chron. 28:9; 1 Kings 3:3. - (source).

Yes, the Bible is explicit no one is in heaven yet:

These were all commended for their faith, yet none of them received what had been promised, since God had planned something better for us so that only together with us would they be made perfect.

Heb 11:39-40

as well as a myriad of threads throughout the Bible, from the firstborn/first/second resurrection, to God's plan of salvation for mankind in Lev 23 Holy Days (not essential for salvation, but definitely beneficial to keep and understand), and more.

No one's dying and going to heaven, or hell, at this point (irregardless of the majority of mainstream Christianity claiming otherwise).

I think there are 2 underlying assumptions in your question. 1. If a person has multiple wives (lives in direct disobedience) or 2. has his heart turned away from God that they are not redeemed.

Things to remember:

1. You will only see human sinners in heaven. Murder (David, Moses, and Paul). Paul participated in the murder of God's people.
2. People are always saved by Grace through faith and not by living according to a moral code of ethics. (all of humanity has failed to lived up to this standard and is not redeemed by this standard) This is why we need Jesus.
3. Obedience is a vehicle to reward or loss of reward, not justification by faith otherwise known as salvation.
4. When a believer turns their heart away from God, God does not renegotiate the promises made at the moment of faith.
5. There are consequences for a believer turning their heart away including physical death.
6. Believers who have their hearts turned away often come back and have their hearts turned back to God.
7. God remains faithful we we are faithless.

"For no Scripture is of any private origin for prophecy never came by the will of man, but holy men of God spoke as they were moved by the Holy Spirit. (2 Peter 1:20-21)

This tells us that Solomon who wrote much of Proverbs, and all of Ecclesiastes and the Song of Solomon was a holy man of God. As such he is in heaven now awaiting the day when he will have a new resurrection body.

Like us, he was a sinner. And like us, all his sins were forgiven because he believed in a coming saviour.

In God's promise to David, Solomon's father, He says,

The Lord declares to you that the Lord himself will establish a house for you: 12When your days are over and you rest with your ancestors, I will raise up your offspring to succeed you, your own flesh and blood, and I will establish his kingdom. 13He is the one who will build a house for my Name, and I will establish the throne of his kingdom forever. 14I will be his father, and he will be my son. When he does wrong, I will punish him with a rod wielded by men, with floggings inflicted by human hands. 15But my love will never be taken away from him, as I took it away from Saul, whom I removed from before you. 16Your house and your kingdom will endure forever before me; your throne will be established forever.’ ”

2 Samuel 7:11-16, emphasis added

You could debate what 'saved' means... But, according to the promise to David, I would say God never turned his face from Solomon.

In that regard, in terms of salvation in the New Testament, as those in the old looked forward in faith to an event they did not know, and those in the New look back in faith to one they do know, I have held and continue to hold that, in this regard, you could say Solomon was 'saved'.

Pertaining to interpretation, while portions of this also point to Christ, I don't see how this part could, since it talks about "doing wrong". In that regard, it seems this portion refers to Solomon literally.