The most common plectrum banjo tuning seems to be CGBD, although DGBD (like a five-string banjo without the fifth string) and "Chicago" tuning (DGBE - like the highest four strings on a guitar) are quite common too.
Although I've never heard of anybody actually using it, the standard Irish tenor tuning GDAE should work well on a plectrum banjo too. In fact most tunings (and corresponding techniques and fingerings) are interchangeable between the plectrum banjo and the tenor banjo. There's only one major exception: Don't try the "standard" CGDA tenor tuning on a plectrum banjo! There's no way you'll get the strings to last with such a high tuning on such a long scale.

The plectrum banjo's relatively mellow tone should be pefrect for Irish/Celtic traditional music. The reason why it's rarely used is probably mostly historical. The instruments the first British banjoists could get hold on were tenor or fivestring banjos and they developed their style to fit these instruments.
There are technical reasons too of course. With the common plectrum banjo tunings you only get a range of an octave and a half before you have to change position. That's not enough for many Irish tunes. As I said you can correct that by tuning it as an Irish tenor banjo, but that would mean some really heavy stretches that limit playing speed and put a strain on the left hand.
There actually is a way to compensate for all the limitations of the plectrum banjo: tune it down to FCAD and put a capo on the second fret - or even down to EBAbDb with a capo on the third fret or EbBbGC/fourth fret capo. That way you effectively end up with a Irish tuned tenor banjo that is slightly harder to keep in tune than usual. But the only reason I can see why anybody would want to do that is if they already have a plectrum banjo they want to use primary for Irish music. Tenors are easier to come by than plectrum banjos, and you loose the plectrum banjo's advantages too if you convert it into a makeshift tenor this way.

Yes, there are plenty of advantages to the plectrum banjo. From a purely tonal point of view, it may well be the banjo type best suited to celtic music. The technical problems may limit your choice of solo pieces to play, but there are still plenty of tunes that fits very well onto its fretboard to choose from. And if you mainly want an instrument to strum chords the plectrum banjo is the perfect choice.

The plectrum banjo

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