Tires with directional arrows usually have a tread that's asymmetric in terms of rotation of the tire about the hub (i.e. the tread appears to "point" forward or backward along the direction of travel of the bike).
Asymmetric tread is designed to improve traction in certain scenarios. For example, they can be designed to encourage water, snow, mud, and other liquid or semi-liquid items to move out from under the tire, improving traction. Mounted incorrectly, they will do the opposite and encourage the same stuff inward toward the center line of the tire.
The direction arrows you mention are to reduce the guess-work and just tell you exactly how to mount the tire to get the tread pointed the right direction. If there is no explicit arrow, look at the tread and see if it has a pattern that "points" in one direction (forward or backward). In every case I've seen, the pattern should point behind towards the back of the bike when it contacts the road, or forward when you look at it from the top.
If there is no direction arrow, and the tire is either slick or has a symmetric tread (doesn't "point" forward or backward), I would say mount it either way. Shouldn't matter.
Also, make sure to look very close for the arrows. Sometimes they're painted onto the tire and easy to see, but other times they're just traced with a raised ridge of tire rubber and that can be hard to spot. Check the entire perimeter of the tire on both sides, in good lighting.
- Did the tire manufacturer include an explicit rotation direction arrow printed or embossed on the tire? If so, follow that.
- If there is no explicit rotation direction arrow included by the manufacturer:
- Is the tread asymmetric in the direction of rotation, i.e. does it appear to "point" forward or backward? If so, mount the tire so the tread is pointing
forward when viewed from the top of the bike.
- Is the tread symmetric (doesn't appear to point forward or backward)? If so, mount it either way, rotation direction doesn't matter.