We authorize the mgmt. co to pay recurring bills - utilities, union, mortgage, payroll. Everything else needs approval by board treasurer or president. In reviewing bills, some may need to be charged back to shareholders or the sponsor for repairs made that were not coop responsibility. I check for that, and also arrears on bills. Our mgmt. co. had failed to pay or correct numerous bills from supers union, they blamed it on staffing changes. If you're really diligent, make a spreadsheet of expected/recurring bills, and see that they are paid when due, and in a $ range similar to previous bills. Check for bills paid that were not yours, but another building. Some bills, (water/sewer, prop. taxes) incur penalties when paid late. This spreadsheet would also help in preparing budgets each year. I also watch for the scheduling and payments of permits, inspections, city/state record filings, etc.
Annual and 5 year elevator inspections, sprinkler and standpipe, oil tank and boiler inspections, local law work - parapet, energy usage (buildings over 50,000 sq feet), corporate registration with NY Dept of State, NYC HPD are some of them. City likes to fine you for missing/late filings/inspections.
What kind of reserve fund do you have? You can direct excess funds in operating account to go to the reserve fund. Reserve fund access should be limited to at least 2 signatures of board members, mgmt. should not have withdrawal access.
The annual budget is an important responsibility, mgmt should present a draft and board reviews and adjusts. This helps the board decide whether to do a maintenance increase to cover expenses. Also, you help determine how to handle the coop abatements and other tax credits and necessary assessments.
You have options as to whom prepares your 1098 forms each year, the accountant or the management co. can both do it. Annual fire safety and window guard notices can be done in house, by mgmt, or by third parties. Do you mail info regularly to owners/residents or slip them under doors, saving postage? If you have any mechanical type of knowledge, meet with contractors yourself to obtain estimates for projects. Get at least two estimates for anything not urgent.
How deeply you want to get involved and how much time you have, there's lots you can do.