He is suffering from a skin disease on his lips – how should he do wudoo’?

Firstly: 

We ask Allaah to heal you and make you well. 

Secondly: 

If your lips are harmed by using water when doing wudoo’,
then you have to wash whatever you can of your face, and the other parts of
the body that are to be washed in wudoo’, and then do tayammum instead of
rinsing the mouth and washing the lips and surrounding area, because Allaah
says (interpretation of the meaning): 

“So keep your duty to Allaah and fear Him as much as you
can”

[al-Taghaabun 64:16] 

You can do tayammum before or after doing wudoo’. 

With regard to not doing wudoo’ 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 wudoo’ 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 wudoo’, 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 wudoo’, and not do tayammum. 

Fataawa Arkaan al-Islam by
Shaykh Ibn ‘Uthaymeen, p. 234. 

Shaykh Ibn Baaz (may Allaah have mercy on him) was asked
about a person who did wudoo’ 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 wudoo’, 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 wudoo’ 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’ Fataawa
al-Shaykh Ibn Baaz, 10/197.