Praise be to Allah.
We ask Allah to heal you and make you well.
If your lips are harmed by using water when doing Wudu, then you have to wash whatever you can of your face, and the other parts of the body that are to be washed in Wudu, and then do tayammum instead of rinsing the mouth and washing the lips and surrounding area, because Allah says (interpretation of the meaning):
“So keep your duty to Allah and fear Him as much as you can” [al-Taghaabun 64:16]
You can do tayammum before or after doing Wudu.
With regard to not doing Wudu at all, and doing tayammum only, that is not permissible, rather in this case you have to combine washing the healthy parts of the body with doing tayammum.
What we have said about Wudu applies also to ghusl. You have to wash your body and whatever you can of your face, as well as doing tayammum.
It says in Zaad al-Mustaqni’: Whoever has been wounded may do tayammum for that part, and wash the rest. End quote.
i.e., if a person has been wounded and cannot wash the site of the wound, then he should do tayammum for that, and wash the rest of his body that is healthy.
The basic principle concerning that is that if a person is wounded or burned in a part of the body that is washed in Wudu, or has some sickness there, such cases fall into four categories:
1 – When it is uncovered and washing it will not harm it. In that case it is obligatory to wash it.
2 – When it is uncovered and washing it will harm it but wiping it will not. In that case he must wipe over it.
3 – When it is uncovered and both washing and wiping will harm it. In that case he should do tayammum for that part, and wash the other parts.
4 – When it is covered with a dressing etc that is essential. In that case he should wipe over the covering, and do the rest of his Wudu, and not do tayammum.
Fatawa Arkaan al-Islam by Shaykh Ibn ‘Uthaymeen, p. 234.
Shaykh Ibn Baaz (may Allah have mercy on him) was asked about a person who did Wudu but his hand was injured and could not be washed with water, and he forgot to do tayammum for it and prayed. He said: If there is a wound in one of the places that must be washed in Wudu, that cannot be washed or wiped because that will lead to the injury being made worse or a delay in recovery, then this person must do tayammum. If a person does Wudu and misses a spot because of a wound, and starts to pray, then during the prayer he remembers that he did not do tayammum, then he should do tayammum and repeat his prayer, because what he did of prayer before doing tayammum is not valid… End quote from Majmoo’ Fatawa al-Shaykh Ibn Baz, 10/197.