Tajweed is an art of reading Quran with a melody, and is a set of rules for non-Arabic readers to reading Quran correctly by using vowelized Arabic text.
The people who claim that Tajweed is required bring the following ayat as their proof:
وَقَالَ الَّذِينَ كَفَرُوا لَوْلَا نُزِّلَ عَلَيْهِ الْقُرْآنُ جُمْلَةً وَاحِدَةً كَذَلِكَ لِنُثَبِّتَ بِهِ فُؤَادَكَ وَرَتَّلْنَاهُ تَرْتِيلًا
And those who disbelieve say, "Why was the Qur'an not revealed to him all at once?" Thus [it is] that We may strengthen thereby your heart. And We have spaced it distinctly.
Furkan 32 (25/32)
This translation is not good, so I need to make some points clear myself. There are three types of "reading" in Quran; "telavat", "qurae" and "tarteel". Telavat is reading a text normally. Qurae is reading it giving your mind completely and trying to understand it. Tarteel is reading a text word-by-word, thinking on each word for a long time, maybe for several minutes. In this ayat, tarteel is used. The bold part of the ayat is translated as "... And tarteel it with a tarteel."
As you see, this ayat has no order of reading Quran in an artistic way with melody. In my opinion, some people love to hear molody, and they want to read Quran with melody too. They think it would be good if they read Quran with melody. But there is no order in Quran to read it with melody. First of all, our main example must be the Rasul Muhammad (pbuh); he never read or made people read Quran with melody in his entire life. Reading Quran in an artistic way is just a bidat that is brought to Islam with some melody-loving people (I'm not saying that loving melody is bad. It is a different subject.).
Allah wants us to read Quran, understand it, and live according to it. Allah and his Rasul does not order us to use Quran as an art tool.
There is another detail about this discussion.
The people who claim that Allah orders us to read Quran with Tajweed also claim that Tajweed is needed for reading Quran correctly.
Their famous example is the similarity between reading of the words "helek (to lose life, to be wasted)", "halak (to shave)", "khalak (to create)".
They say that, if non-Arabic readers don't read Quran according to Tajweed rules, they misread words like these, and they misread the entire Quran at all.
Misreading of non-Arabic readers?
They are already not understanding a word they read. What is the importance of correctly reading a text that you don't understand? The readers that know Arabic are already know how to read and they are correctly reading Quran.