In the Hindu religion, visiting a temple without ablution (cleaning dirt) is said to be sin. One should always bathe before visiting the temple. But in Srisailam, the story is completely different. A person should enter without cleaning the dirt from their body. If he/she does bathe to clear the dirt it is said to be a sin. Why is this so? What is the story behind this?
The general practice at Srisailam is to first bathe in the Patalaganga (Krishna river), and walk from there to the Temple, for which a special walkway exists. It is part of the basic etiquette of any temple to bathe before entering the shrine, as reintererated in the Devasthanam's website.
The standard practice of visiting any sacred kshetra or dham is to first perform tarpan to your ancestors on the first day, after which you may go take darshan. Most Shiva temples (and most Jyotirlingas) allow entry in the garbha griha, and actually touch the linga (sparsha darshanam). Lord Shiva is not contaminated by anything, and as the Srikalahasti temple shows, not even by eclipses.
The dhuli darshan is a practice popular among devotees from Karnataka. In south India, the month of kartik is sacred to Shiva, and many devotees would walk all the way to Srisailam. For these hardcore devotees, performing tarpan, walking to the Patalaganga and back seemed superfluous, and would directly take Darshan first. This is the dhuli darshan performed by some, not all, and particularly during kartik month. source
Dhooli/Dhuli means Dirt, it is not considered as a sin if pilgrims bathes and enters sanctum sanctorum.
The temple allows devotees to worship the Shiva Lingam without bathing as it is said that lord doesn't judge his devotees by his physical impurities but by his intrinsic spirituality.
It is said that at Sreesailam,
Lord Shiva is manifested as a father and when his children (devotees) visit him, he is eager to see them upfront without any delay and grant all the wishes.
Though, some pilgrims do bathe at Patalaganga, which is about 3KM away from Mallikarjuna temple.
As you commented and as I stated before, its more of a belief or custom, rather than a story behind that.
In Srisailam there is a peculiar custom known as 'dhuli darshan.' This is mostly followed by pilgrims who are from Karnataka. At the time of the dasara festival many pilgrims from Karnataka start pilgrimate towards Srisailam on foot.
After climbing the footstairs of the hill the pilgrims straightaway walk into the sanctum sanstorum of the Mallikarjunaswami temple without taking bath at Krishna Ghat for having darshna.
Reference: Encyclopedia of the Śaivism