Resorting to Tayammum if One Has Eczema
Question

I have eczema which was under control until I started doing Wudu 5 times a day. I have tried everything I can think of, but nothing is helping. What can I do if Wudu is causing very bad rashes?
Praise be to Allah.Allah says (interpretation of the meaning): 
“Allah burdens not a person beyond his scope.” [Al-Baqarah 2:286]
Abu Hurayrah (may Allah be pleased with him) narrated that the Prophet (peace and blessings of Allah be upon him) said: “… if I forbid you from doing  something, avoid it, and if I command you to do something, do as much as you can.” (Narrated by Al-Bukhari, 6858; Muslim, 1337) 
Although Allah has clearly commanded us to do Wudu with water, He exempted the sick from having to do Wudu with water and has allowed them to do Tayammum . Allah says (interpretation of the meaning): 
“O you who believe! Approach not As-Salah (the prayer) when you are in a drunken state until you know (the meaning) of what you utter, nor when you are in a state of Janabah (i.e. in a state of sexual impurity and have not yet taken a bath), except when travelling on the road (without enough water, or just passing through a mosque), till you wash your whole body. And if you are ill, or on a journey, or one of you comes after answering the call of nature, or you have been in contact with women (by sexual relations) and you find no water, do Tayammum with clean earth and rub therewith your faces and hands (Tayammum). Truly, Allah is Ever Oft-Pardoning, Oft-Forgiving.” [Al-Nisa’ 4:43]
Shaykh Muhammad Al-Salih Al-‘Uthaymin (may Allah have mercy on him) said: 
“Or if he is afraid of using it or if looking for it will cause him physical harm: 
If he will be physically harmed by using water and will become sick, then this comes under the general meaning of the words (interpretation of the meaning): 
“But if you are ill or on a journey.” [Al-Maidah 5:6]
For example, if there are wounds or sores on the parts of the body that are washed during Wudu, or if there are wounds or sores on all parts of the body, in the case of Ghusl (ritual bath), and there is the fear of physical harm, then he may do Tayammum.” (Al-Sharh Al-Mumti’, 1/378, 379; Ibn Al-Jawzi edition) 
But if he can put water on his skin without affecting it, then he should do so, and he does not have to rub that part of the body or go to extremes in washing it. 
Shaykh Muhammad Al-Salih Al-‘Uthaymin (may Allah have mercy on him) said: 
“Similarly he does not have to go to extremes in rinsing his nose if he has a problem in the sinuses, because in that case the water may settle there and cause an unpleasant smell and make him sick, or it may cause him some harm. It should be said to such a person: it is sufficient for you to rinse your nose until the water enters the nostrils.” (Al-Sharh Al-Mumti’, 1/210, Ibn Al-Jawzi edition) 
He also said: 
“If a person has a sinus problem, or if going to extremes in rinsing the nose will make the water reach the sinuses and cause him pain, or if the water will turn rotten and turn to pus and so on, in this case we say to him: Do not go to extremes in rinsing your nose, so as to ward off harm from yourself.” (Al-Sharh Al-Mumti’, 1/172. Ibn Al-Jawzi edition) 
If Wudu will harm your skin or delay your recovery, then you should do Tayammum , and there is no sin on you. 
And Allah knows best.