The airline performance department should be checking NOTAMs and assessing whether any obstacles require recalculation of the RTOWs.
I've flown with two divisions of the same company that use the same performance department. In the first division we would get a "NOTAM watch" published twice weekly which listed all NOTAMed obstacles for our destinations and how they affected our RTOW charts. You then go through the NOTAMs and match them up with the NOTAM watch. If anything wasn't on the NOTAM watch you needed to have a think about it or get in touch with the performance department. Typically you could use a bit of logic to tick off obstacles as N/A. For example any obstacles behind the the start of the take-off roll wouldn't be an issue, likewise an obstacle that is lower and further down the take-off splay than another obstacle that has been assessed as not affecting the RTOW would be ok.
Bizarrely I now fly for this other division of the company and we get no information at all about obstacles, so there is no way as a captain to easily assess which obstacles are ok and which aren't. The crazy thing is that the NOTAM watch I used to get was not specific to that part of the company, it had all of the company routes / aircraft types on it, so the information is out there, we just don't get it.
When I've enquired about it I'm told that we just trust that the performance department are doing their job and if any significant obstacles are NOTAMed we'll get some new RTOWs.