If a kosher pot is washed up with cold water and washing up liquid, and a washing up pad that has been used to wash non-kosher pots - what is the status of the pot?
The classical case is one of washing a dairy pot with a meat sponge or vice versa. If the pot and cleaning water are cold, the pot is fine (but needs to be cleaned well with an appropriate sponge) and the sponge should be thrown away (as it is too difficult to clean). See sources below.
If the pot had actually fleishig on it, then the sponge should be thrown out because it is too hard to clean. If it was a clean pot and it was only used for pareve, then the sponge is alright.
Regarding the pot: If it was washed with cold water or even warm water, that wasn’t yad soledes bo, 110 F, then it is fine. Just wash it off with soap to get off anything the sponge might have left on it, and it is alright.
Halachafortoday writes (under Jan 14, 2010)
If one class of dishes was mistakenly washed with a sponge of the other class, in most cases the dishes do not become non Kosher, as the water is usually not hot enough. However, the utensil should immediately be re-rinsed with cold water and with the proper cleaning apparatus. If the dish in question is glass, there is room for even more leniency. As always, whenever in doubt, a Rav must be consulted.