Why does a servant feel that God is far from him when supplicating, yet close to him when sinning?

It is not possible to definitively interpret this feeling. However, among the ill effects of sins are the estrangement between a servant and his Lord, and entertaining ill thoughts about Allah, as Ibn al-Qayyim mentioned. The questioner's feeling of Allah being distant when making supplication might be due to sins from which he has not repented with sincere repentance. Meanwhile, his feeling of Allah being close when committing a sin might be from the embellishments of Satan, or because sins can lead to inner forms of servitude like remorse and fear, or outward ones like weeping, seeking forgiveness, and following them with good deeds. The real benefit lies in the praiseworthy effects that follow the sin, not in the act of sin itself. Repentance makes the heart closer to Allah and increases its love and longing for Him.