Faith consists of:
1) Belief in the heart.
2) Actions of the Tongue.
3) Actions of the Limbs.
If you believe in your heart, you have to testify that with your tongue (i.e. pronounce Laa illaaha illAllaah, Muhammadun Rasool Allaah [there is none worthy of worship except God/Allah, and Muhammad (peace be upon him is His Messenger.]
This testification is then an action of the tongue, and required in order for one to be a believer and muslim.
As brother Muhammad explained;
Islam is based on five pillars which the Prophet Muhammad (peace and blessings of Allaah be upon him) has explained for us, when he said:
“Islam is built upon five [pillars]: the testimony that there is no god but Allaah and that Muhammad is the Messenger of Allaah; establishing regular prayer (salaah); paying the zakaah; Hajj (pilgrimage) and fasting Ramadaan.”(Agreed upon; narrated by al-Bukhaari, no. 8)
The above 5 pillars are part of Islam, and for one to be a Muslim - they need to stick to them. The proof for that is in the hadith when a bedouin came to the Messenger of Allah (peace be upon him):
Sahih Muslim Book 001 [Kitab ul Eman (Book of Faith), Number 0014:]
It is reported on the authority of Abu Huraira that a bedouin came to the Messenger of Allah (may peace be upon him) and said: Messenger of Allah, direct me to a deed by which I may be entitled to enter Paradise.
Upon this he (the Holy Prophet) remarked: You worship Allah and never associate anything with Him, establish the obligatory prayer, and pay the Zakat which is incumbent upon you, and observe the fast of Ramadan. He (the bedouin) said: By Him in Whose hand is my life, I will never add anything to it, nor will I diminish anything from it. When he (the bedouin) turned his back, the Prophet (may peace be upon him) said: He who is pleased to see a man from the dwellers of Paradise should catch a glimpse of him.
So we see from the narration above that even if someone sticks to the basic minimum i.e. the 5 pillars, then they will enter Paradise. If they were a sinner, then they may be punished or forgiven on the Day of Judgment by Allah.
The first stage a person goes through when becoming Muslim is to submit themselves, as they gradually increase in faith - they become a Mu'min, i.e. they gain firmer Emaan [conviction in faith], then the highest rank is to have Ihsaan - every single aspect of your life involves you having a stronger relationship with God/Allah.
To answer your question, we know that if someone is to believe in Allaah, then they are ordered to believe in His Messengers', and what they came with. Allah revealed the Qur'an to His final Messenger (peace be upon him) - the Qur'an is the speech of Allah.
How can someone claim to be a believer if they reject Allah in the first place? How can they believe in Allah's Messengers' if they deny the fact that Allah exists? If they reject Allah, they reject His Messengers', His Angels, His Books, the Hereafter, the Day of Judgement, and the concept of destiny. These beliefs are vital in order for one to be a true believer.
Some people claim that belief on the tongue without belief in the heart is sufficient, yet this is a clear lie since Allah says (translation of the meaning):
When the hypocrites come to you (O Muhammad - peace be upon him),they say: "We bear witness that you are indeed the Messenger of Allah."
Allah knows that you are indeed His Messenger and Allah bears witness that the hypocrites are liars indeed.
[Qur'an The Hypocrites [Surah Al Munaafiqun] 63: 1]
The above verse then proves that simply to say that 'i believe' with ones tongue isn't sufficient, rather that belief has to settle in the heart aswell as be testified on the tongue, and proven through actions. All of these 3 act together and are dependant upon each other.
If someone says the testimony of faith with their tongue, and shows it through their actions, while rejecting Allah and His Messenger in their heart. They may be termed as hypocrites. But as Muslims, we are obliged to accept them as part of the Ummah [Muslim community.] Then on the Day of Judgement, their Judgement is with Allah. And on that day, no-one will be dealt with unjustly.
And Allaah knows best.
Peace.