Wikipedia puts the Tarantula Nebula at 49 kpc (160,000 ly), citing
this paper, and the LMC at 50 kpc (163,000 ly), citing
this paper. I'm not sure if the first paper's distance is supposed to be the distance to the nebula or to the galaxy, but I think the discrepancy is probably just due to the first paper being older. I vote same distance, or unknown.
Added after 5 hours 54 minutes:Turns out I was right that the difference is only due to one estimate being older;
this is the original source for the 49 kpc distance, and it says (my bold):
[...]we obtained a distance modulus of (m-M)0 = 18.45 ± 0.15 for 30 Doradus in excellent agreement with the best estimate for distance to the LMC, (m-M)0 = 18.50.
Added after 30 minutes 9 seconds:Just to be clear: the 49 kpc distance to the Tarantula Nebula is from a 1999 paper that says it is "in excellent agreement" with the distance to the LMC. This has been superseded by a more accurate distance to the LMC of 50 kpc from 2013 (which is also very close to the distance in Celestia). No paper actually says that the Tarantula Nebula is closer than the LMC's central bar, and that conclusion can only be reached by assuming that an older distance estimate is just as accurate as a newer one.