This is one of several reasons why I personally do not like inkjet media. The disc could sit out for a month -- a year -- and it would still smear off when rubbed with any amount of force. The oils in your hands are enough to damage the surface of the disc. Sometimes humidity in the air alone is enough to cause damage.
Some discs also hold ink better than others. However, this topic is constantly debated, as to which disc is "best" for ink use. I tend to prefer Verbatim's inkjet surface for those few times when I am forced to print.
My quick rant:
It's a lot of effort to spend on creating artwork, costs of printers, extra discs expenses -- all for something that gets maybe 2 seconds of viewing time. Effort is far better spent on DVD menus and DVD cases, as both of those get several minutes of total viewing time, compared to a couple of seconds as the disc goes from case to player. It seems like such a waste of time and resources to print on a disc. /End rant
The dried-but-smearing ink should not do any harm to the DVD player, no. There is nothing to "clog up" -- it's just a smear on the disc surface.
Some folks suggest various spray-on coatings
, as used by artists to "cure" canvas paintings. They swear by its success, but I don't know if I would suggest it. It might be worth investigating, if you're determined to print inkjet media, and are concerned about how easily it smears.