I feel guilty for derailing the thread. Back to serious advice -
- At some temples you'll see a large incense burner in the temple grounds (exterior) - with people pulling smoke over themselves as a purification ritual. It will have clouds of incense around it but you can usually get past without going too close, and there isn't so much incense inside.
- Not all temples are alike. Some are like parks with clean air (e.g. Kinkakuji in Kyoto) and others like crowded bazaars with competing smells from all sides (e.g. Sensoji in Tokyo). Little neighborhood temples can be very tranquil spaces with not much going on.
- When funerals and other rites are being performed, quite a lot of incense is burned, so you might want to avoid entering rooms where you see people participating in a ceremony.
- Buy a face mask and some antihistamines. Wearing a mask is not seen as eccentric in Japan.
- Carry a spare jacket or sweater, so that you can change your outer layer after leaving a smoky place.

Even if they don't have a reaction to incense, many tourists find that they quickly tire of visiting temples. Fortunately, there are many other sites to visit, and for the most part, Japanese people do not use strong scents in buildings or on themselves.