Yes, you can get illnesses from swallowing semen. It is possible to get throat infections of gonorrhea and chlamydia if you swallow infected semen. Semen can contain HIV and Hepatitis B, too. And even if you didn't swallow the semen, just the oral friction contact between you and his penis can transmit syphilis and herpes (with the resulting sores appearing on or around your mouth). Many studies have also shown that HPV, the virus that causes cervical cancer and genital warts, is linked to throat cancers in people who regularly engaged in unprotected oral sex.
Oral sex is not a perfectly safe activity. If you both haven't been tested for STDs lately, you shouldn't assume you are both clear of infections. To enjoy safer oral sex (or safer sex in general), get tested for STDs every 6 months, use a flavored condom when you give him oral sex, and let him cover your vulva with a sheet of plastic wrap when he performs oral sex on you. Some people might complain that the latex and plastic lessen sensations, but you can solve that issue with a dab of water-based lube on the genitals.