No. Someone else's urine will not change your DNA. In fact, there is no substance you can ingest that will change your body's DNA.
I also believe you are overdoing it by saying so sweepingly that urine "contains" DNA.
As we have said, even if urine was loaded with DNA, it would not affect your DNA, but also, urine is not a source of DNA.
Here are from a couple of websites dealing with trying to get DNA from a urine sample for purposes of police work:
"Urine itself does not contain DNA, but it may contain epithelial cells, which do contain DNA. Most healthy individuals, however, do not excrete epithelial cells in their urine."
And from another: "Urine is not considered an ideal source of DNA due to the low concentration of nucleated cells present in human urine. The nucleated cells found in urine are typically white blood cells and epithelial cells. There are large differences between the amount of epithelial cells present in male and female urine. Females are more likely to have a higher number of epithelial cells present in their urine, vaginal cells being the chief contributor."
If anything physical could come from accidentally drinking urine, it might be from bacteria, but even that is unlikely. From a website asking that question: "Urine is sterile because it contains no living organisms, unless the person that produces is unlucky enough to have a urinary tract or bladder infection. There are less bacteria in urine than in tap water, for example."
A problem from drinking urine would be psychological more than physical.