It is well known that Vedanta Sutras prohibit Shudras from Jnana by saying Smritis prohibit them from having Upanayana. But Bhagavad Gita is a book which contains Vedantic statements. As Vedanta Sutras prohibit Shudras from acquiring Vedanta knowledge, can't they read the Bhagavad Gita?
There are no specific restrictions for reading and understanding Bhagavad Gita based on castes, differentiation apart from two cases, where Krishna himself emphasizes on to whom this Gita knowledge should not be spoken and to whom this knowledge shall be revealed
BG 18.67: This instruction should never be explained to those who are not austere or to those who are not devoted. It should also not be spoken to those who are averse to listening (to spiritual topics), and especially not to those who are envious of me.
BG 18.68: Those, who teach this most confidential knowledge amongst my devotees, perform the greatest act of love. They will come to me without doubt.
Thus, from these verses, it is clear that, those who teach his Gita knowledge to his devotees, certainly reaches Krishna's abode which he himself declares and also the fact that, Sri Krishna makes it clear that this knowledge should not be explained to non-devotees, and non-believers of supreme because they may become envious of Krishna and misunderstand the teachings of Gita and think Krishna proudly spoke of himself and wanted Arjuna to praise him always as he used many times "I". So real meaning and teachings of Gita wont go into minds of non devotees.
Mahabharata is composed by Vyasadeva so that all classes of people will benefit be it sudras, fallen brahmanas etc.. So they can definitely read Bhagavad Gita which is a part of Mahabharata.
strī-śūdra-dvijabandhūnāṁ trayī na śruti-gocarā karma-śreyasi mūḍhānāṁ śreya evaṁ bhaved iha iti bhāratam ākhyānaṁ kṛpayā muninā kṛtam
SB 1.4.25: Out of compassion, the great sage thought it wise that this would enable men to achieve the ultimate goal of life. Thus he compiled the great historical narration called the Mahabharata for women, laborers and friends of the twice-born.
If for Mahabharata (of which Gita is a part) it is said that it can be recited by everyone then "Yes" will be the answer. But have a look at the following verses from Vyasa Smriti:
Brahmanas, Kshatriyas, and Vaishyas are called the twice-born. Their first births take place when they are delivered of their mother's womb ; their second, when they duly accept the Gayatri Mantra from their preceptors (21).
Thus made twice-born, and free from all other faults, they become entitled to study the Vedas, Smritis and Puranas (22).
According to 22, it's the Vedic initiation that makes one qualified to read even the Smritis and the Puranas. So, we can assume that similar requirements might be there for reading the Itihasas too. Because, Itihasa-Purana are always mentioned in tandem.
Also, note that this is somewhat an indirect approach, as ideally one should see what Mahabharata itself( or Gita) has to say on the topic.