Bobby, when I'm analyzing properties, I do consider properties with 0 or even slightly negative cash flow. This CF is after all expenses, including 15-20% reserves for vacancy, maintenance and capital improvements.
If the CF is negative, then I would need to set aside $X for reserve. The reason I allow this in my analyses is because setting aside $4000, for example, to cover reserves until my rent increases catch up is cheaper than going with the lower LTV loan.
I also will try to pull as much money out of the deal as possible. The reason for this is the ROI on the appreciation and mortgage paydown is multiplied by the amount of leverage being used. For example, let's say appreciation is 3% and mortgage paydown that year is 2%, and I only have 10% of my money in the deal. The ROI is 5% x 10 = 50%. If I left 20% of money in the deal, the ROI is 5% x 5 = 25%.
In my analyses, the ROI of the appreciation + mortgage paydown (i.e. amortization) outweighs the ROI I'm able to get on the CF using a lower LTV refi loan.
With that said, I would not consider going with 0 or negative CF if you don't have any enough money set aside to cover mortgage payments for roughly 3-6 months. If for some reason you don't have tenants for an extended period, you need to be able to still make those payments. So, always consider your personal situation when deciding your investing strategy.