The Mitzvah to Believe and the Principles of Faith
The Rambam has 13 principles of faith that a person must believe in. Part of the definition of Orthodoxy is having belief in certain fundamentals of faith. These principles are necessary and are building blocks for most other hashkafa's and values in Torah.
- Existence of God
- Oneness of God
- Non-physicality of God
- God is the creator of everything
- There is prophecy
- Moshe's prophecy was unique
- Torah is divine
- Torah doesn't change
- God is omniscient
- There is reward and punishment
- Mashiach will come
- The dead will be revived
Obligation to Believe
- There is a mitzvah to believe in Hashem.
- There is a mitzvah to believe that Hashem is one.
- One may not believe in any other God besides Hashem.
- There is a mitzvah to believe that there will be an ultimate redemption brought about by Mashiach.
Prohibition to Stray after other Beliefs
- There is a constant Biblical prohibition to stray after thoughts that are against our belief.
Proving the Existence of God
Ramification of Disbelief
- A person who does not believe in any of the 13 principles of faith loses their share in Olam Haba. 
Causing others not to believe
- Some say that it is forbidden to be a librarian in a secular library because in doing so you'll have to give books which speak of heretical ideas to anyone who wants them, which is a violation of Lifnei Iver. 
- Pirush Mishnayot (Sanhedrin 10:1)
- Shemot 20:2, Rambam (Sefer HaMitzvot, Aseh n. 1), Sefer HaChinuch (n. 25), Smag (Aseh n. 1), Chaye Adam 1:5, Biur Halacha (1 s.v. Hu)
- Devarim 6:4, Rambam (Sefer HaMitzvot, Aseh n. 2), Sefer HaChinuch (n. 417), Smag (Aseh n. 2), Chaye Adam 1:5, Biur Halacha (1 s.v. Hu)
- Shemot 20:3, Rambam (Sefer HaMitzvot, Lavin n. 1), Sefer HaChinuch (n. 26), Smag (Lavin n. 1), Chaye Adam 1:5, Biur Halacha (1 s.v. Hu)
- Smak (Mitzvah n. 1)
- Brachot 12b derives from the pasuk Bamidbar 15:39 that there is a prohibition to stray after anti-Torah beliefs.
- Pirush Mishnayot (Sanhedrin 10:1), Rambam Teshuva 3:6-8
- Yalkut Yosef YD 157:7