The use of "falls on" indicates that it seems a bit of a "coincidence" or an interesting fact to you. Valentine's Day is always February 14 and it happens to fall on a Thursday this year. In the US, Thanksgiving is always the fourth Thursday in November and it happens to fall on November 28 this year.
The funeral is planned for a certain day of the week or a certain date so it might be possible to say it falls on whichever of the two was not specifically chosen, however since funerals generally are within a week of the death, the day and date are both closely associated - on Monday the 6th, it doesn't seem coincidental or interesting that the 7th is a Tuesday or that Tuesday is the 7th.
I wouldn't use "falls" for a funeral, as if it were happening year after year on a particular date. I might say, "Since the funeral istaking placeon Thursday, I won't be able to attend, because . . . ". (alternatives: will take place on, is planned for)