In a word: no. While Japan still lags behind the UK and US in gay and transgender issues, there are no legal restraints to same-sex sexual activities of either gender.
We're proud to have arranged many trips for gay couples and have had no problem booking double bedrooms regardless of the travellers' gender. Public displays of affection are rare between same-sex couples in Japan, but the same can be said for heterosexual couples. You may raise a few eyebrows if you hold hands in the countryside, but that's probably about all.