Urine may be a bit smelly but apart from that there is no reason not to use it! As long as you have a pair of gloves on and rinse the clothes with tap water after it should be fine. We are all very delicate these days in our highly protected and politically correct lives, why not actually do something a bit different that is also effective!
Just because the Romans used urine to clean clothes does not mean modern people should. They also used a number of things containing copious amounts of lead and we all know that lead is harmful to the human brain. If their is an EMP, a super volcano eruption or other calamitous event we should keep the technique in our mind; however, barring that budget savvy people should resort to saving money elsewhere.
The Romans also used public toilets with shared toilet brushes to wipe themselves instead of toilet paper. Now that we have access to soaps and detergents that do not pollute the environment,there is no reason to walk around smelling like a urinal. Don't you wash kids clothes and sheets when they urinate on or in them? There is a reason why other cultures never adopted the practice of washing clothes with urine.
The Romans did what they could with what they understood. Modern Americans, budget-minded or not, would do well to take note of the greater depth of knowledge we have regarding the cleanliness of urine and the other low-cost options available for cleaning products. Inexpensive items like baking soda cost very, very little and are much more effective at actually cleaning.