You can always try putting this code in your main css without modifying the bootstrap file what is most recommended so you keep consistency if in a future you update the bootstrap file.
.tooltip-inner {
background-color: #FF0000;
}
.tooltip.right .tooltip-arrow {
border-right: 5px solid #FF0000;
}
Notice that this example is for a right tooltip. The tooltip-inner property changes the tooltip BG color, the other one changes the arrow color.