Short answer
No. A Muslim man's awrah is from the navel to the knees, so he does not cover his face, and a man wearing a burqa or a woman's veil would be imitating women, which is forbidden. Men may wrap a scarf over the face against dust, cold or sun, but covering the mouth in prayer without need is disliked.
Men are not required to cover their faces, and a man wearing a burqa or a woman's face veil is imitating women, which is haram. Covering the mouth and nose during salah without need is makruh.
The four schools agree on this.
Explanation
The burqa and niqab are garments for women, worn as part of hijab; see the hijab guide. For men, the area that must be covered is from the navel to the knees, and the four schools agree that a man's face is not part of it. The Prophet ﷺ cursed men who imitate women, so a man wearing a woman's face veil would be sinning, not being extra modest.
Some men do cover their faces for practical reasons. Tuareg men in the Sahara wear a long veil, and men in the Gulf and in deserts wrap a shemagh or ghutra across the face in sandstorms and sun. These are traditional or practical, not a religious requirement, and they are allowed.
- In prayer: the Prophet ﷺ forbade a man from covering his mouth in salah, and the Hanafi texts call it makruh. A medical mask worn because of illness or infection is a need, and there is no dislike then.
- In ihram: men may not cover their faces in the Hanafi school; see can you wear a face mask in ihram.
- Head: many Muslim men cover their heads with a cap or turban, which is a sunnah of habit; see do Muslim men have to cover their heads.
For the whole men's dress code, see the dress code for men.
The evidence
Quran, Surah Al-A'raaf 7:26
يَٰبَنِىٓ ءَادَمَ قَدْ أَنزَلْنَا عَلَيْكُمْ لِبَاسًۭا يُوَٰرِى سَوْءَٰتِكُمْ وَرِيشًۭا ۖ وَلِبَاسُ ٱلتَّقْوَىٰ ذَٰلِكَ خَيْرٌۭ ۚ ذَٰلِكَ مِنْ ءَايَٰتِ ٱللَّهِ لَعَلَّهُمْ يَذَّكَّرُونَ ﴿٢٦﴾
O children of Adam, We have bestowed upon you clothing to conceal your private parts and as adornment. But the clothing of righteousness - that is best. That is from the signs of Allah that perhaps they will remember.
Clothing is for covering the awrah and for adornment.
Sunan Abi Dawud 643
Grading to be checked... نهى عن السدل في الصلاة وأن يغطي الرجل فاه ...
"He forbade letting the garment hang loosely in prayer and forbade a man from covering his mouth."
The basis for disliking a covered mouth in salah.
Sahih al-Bukhari 5885
Sahih... المتشبهين من الرجال بالنساء والمتشبهات من النساء بالرجال ...
"The Messenger of Allah cursed men who imitate women and women who imitate men."
Ibn Abbas's report that imitating the other sex in dress and grooming is a major sin.
Where the ruling is recorded
- al-Durr al-Mukhtar with Radd al-Muhtar, Kitab al-Salah, chapter on what is disliked in prayer. Covering the nose and mouth in prayer is disliked.
- al-Hidaya, Kitab al-Salah, chapter on the conditions of prayer. A man's awrah is from below the navel to the knee.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.