Is it permissible to fast on Tuesdays and Wednesdays with the intention of worship and drawing closer to Allah?

There is no evidence in the Sunnah indicating the recommendation of fasting specifically on Tuesdays and Wednesdays. However, fasting on these days is permissible and carries the general virtue of fasting.

Shaykh al-Uthaymeen, may Allah have mercy on him, said: "Fasting on these two days is not a specific Sunnah, but it is recommended to increase fasting in general. Fasting on them is not disliked. In summary, fasting on Tuesdays and Wednesdays is permissible; singling them out for fasting is neither Sunnah nor disliked. In contrast, it is disliked to single out Friday, Saturday, and Sunday for fasting, though it is permissible to combine them with other days. As for Monday and Thursday, fasting on them is a Sunnah."