doesn't sound like tendonitis ... may be a heel bursa caused by the shoe rubbing. Still need to ice soundly after exercise ( 20-30 min ) Try reducing friction on the heel ( shiny plaster on heel and soap it to reduce friction, " second skin " or two pairs of thin socks.)
as to ignoring it... in my experience that tends to lead to bigger problems further down the road. Often early diagnosis and a bit of treatment will put things right quickly.
I should talk...now into third week of no running after trying to race with tendonitis, you'd think i'd know better at my age. I
do but impatient , and they were GOOD races.