I have a question to you :)
Assuming we have a rider moving on a horse. He needs to get to destination ASAP, so he wants his horse to force march today.
Reading Handle Animal skill, I've noticed that this is being handled via Push Animal option and it has DC 25.
So, assume that our rider has Handle Animal bonus of +5 and he wants his horse to force march for 3 hours.
How should I handle this in-game? Every hour he should spend 2 minutes to "take 20" (retry is allowed for this skill)? Or not?
the movement rules say that you only need to make 1 check every hour when doing a forced march. The initial check as you stated is a DC 25. Assuming the PC passes this check after 1 hour the horse will take 1d6 lethal damage and become fatigued. This means that now it can't run or charge and it takes a -2 to str and dex (reducing its carrying capacity).
At this point the PC can make another check but the DC increases by 2 since the mount is injured. Meaning its now a DC 27. Assuming the character was able to pass this check after 1 hour the horse will take an additional 1d6 damage and becomes exhausted. This means that now it can't run or charge, it takes a -6 to str and dex and it only moves at half speed.
In theory the PC could make another check again at a DC 27 to push the animal further, but will only make half as much progress for the time spent. After each hour the horse would take another 1d6 damage at the same penalties until the horse dies from the lethal damage.
In your specific case since the character only adds +5 to the check they will have a hard time. They could take 10 but that won't be enough to pass the check. Taking 20 by RAW would mean they pass the check after 20 hours of trying. Since this isn't reasonable, you could make each check only take 1 minute. This means that it takes 20 minutes to get started. After 1 hour of travel the horse would take damage and become fatigued. The extra +2 to the DC is because it's injured. If they have some way to heal the horse, they could then spend another 20 minutes to get it moving again. If they don't have any way to heal it, their skill is too low to force the mount to move any further as they can't pass the DC 27 even taking 20 (a roll of 20 on skill checks does not denote automatic success).
The characters can avoid trying to move at half speed with an exhausted mount by allowing it to rest for 1 hour. This reduces it back to only being fatigued. It takes 8 hours of continuous rest to remove the fatigued condition.
Forced march rules used on mounts really only works for the last 1 or 2 hours of the day, and results in a wounded and slowed mount. If you really need to make more daily progress, you need to have fresh mounts waiting for you at staging points, or some magical way to travel without normal fatigue-prone mounts.