He is worried that he may have caused harm to another person
Question

Perhaps the question I am going to ask may seem very strange, but what is making me ask it is feelings of regret and worry. What happened is: one day I wanted to fix the tires on my car with the mechanic. On my way to the repair shop, there was a puddle of water in my way, and I did not know whether it was rainwater or wastewater (sewage). I drove through it, and after I reached the repair shop, I was worried about my own safety, and I did not touch the tires of the car, because I know that wastewater and dirty things on the ground may carry dangerous bacteria or viruses that can cause diseases that are fatal to humans. But at the same time I was careless and I did not tell the mechanic to wear something protective before touching the tires, so that his hands would not be contaminated.
My question is: in the event – Allah forbid – that something bad happens to this man because of me, and he dies, would this be regarded as killing – Allah forbid – or accidental killing? What exactly would my situation be, according to sharee‘ah?
Praise be to Allah.This anxiety that you are feeling and these worries that are making you think that you caused harm to someone else is based on nothing but doubts and waswas (compulsive insinuating thoughts). As for the water you drove through, you do not know what its condition was at all, and the basic principle is that mud and water in the streets is pure, unless it is known that it is impure water. 
Moreover, the wheels of your car drove over the ground after that, which usually helps to remove dirt and clean off whatever got stuck to the wheels before that. 
But regardless of all that, such dirt is visible to the eye, so if there was anything that needed cleaning or precautions, this man would have taken care of it, because this is his profession. There would not be anything strange on your car that he would not notice; in fact he is most likely used to that, and is more knowledgeable about that which requires protection and that which does not in such cases. 
Undoubtedly what you mentioned about the possibility of harm to the mechanic is no more than waswas and whispers from the Shaytan, which he is instilling in your mind so that you will remain anxious and worried, and upset about something that is not going to have any effect. Just as the Shaytan seeks to tempt people and divert them from doing good by means of whims and desires, and specious arguments, he also tries to keep them from doing righteous deeds and beneficial acts by means of waswas and sadness. 
Feeling sad for something that you could not make happen or that you could not do anything about if it did happen after the fact is going to cause you to suffer from waswas and compulsive thoughts, and it will distract you from focusing on that which is in your own best interests and make you feel weak and exhausted. That is something that is not right and is blameworthy. 
Hence the Prophet (blessings and peace of Allah be upon him) used to seek refuge with Allah from worry and sorrow, as is narrated in the hadeeth of Anas ibn Malik (may Allah be pleased with him) who said: The Prophet (blessings and peace of Allah be upon him) used to say: ““Allahumma inni a’oodhu bika min al-hammi wa’l-hazani wa’l-‘ajzi wa’l-kasali wa’l-jubni wa’l-bukhli wa dala‘ id-dayn wa ghalabat ir-rijal (O Allah, I seek refuge with You from worry and grief, and I seek refuge with You from incapacity and laziness, and I seek refuge with You from cowardice and miserliness, and I seek refuge with You from the stress of debt and from being overcome by men).””Narrated by al-Bukhari (6369). 
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: 
With regard to “grief” or sadness, it is not enjoined by Allah or His Messenger; rather it is disallowed in some cases, even if it has to do with matters of religion, as in the verses in which Allah, may He be exalted, says (interpretation of the meaning):
So do not weaken and do not grieve, and you will be superior if you are [true] believers. (Al ‘Imran 3:139)
And be patient, [O Muhammad], and your patience is not but through Allah . And do not grieve over them and do not be in distress over what they conspire. (an-Nahl 16:127)
and he said to his companion, "Do not grieve; indeed Allah is with us." (At-Tawbah 9:40)
And let not their speech grieve you. (Yoonus 10:65)
In order that you not despair over what has eluded you and not exult [in pride] over what He has given you. (al-Hadeed 57:23)
And there are many similar verses. 
That is because it does not bring any benefit or ward off any harm, so it serves no interest, and that which serves no interest cannot be enjoined by Allah… But if it leads to weakness (of faith) in the heart and distracts one from doing that which Allah and His Messenger have enjoined, then it is blameworthy.End quote from Majmoo‘ al-Fatawa, 10/16-17 
What you must do is focus on that which will benefit you and ward off from you this waswas, compulsive thoughts, doubts and illusions. 
Please see the answer to question no.  102851
And Allah knows best.