Front wheel lifting on climbs on my Mountain Bike
I had the same problem on a regular non-suspension bike on the road, if the slope is steep enough.  The main cause is technique, and its exacerbated by your suspensions.

As you push down on the leading pedal the bike wants to rotate the other way, like you're lifting the handlebars and pulling a wheelie.  Your suspension is probably acting as an amplifier for this.

This leads to decreased traction on the front wheel, so turning becomes interesting.

Main Solution is to pedal more steadily in a circle, rather than stomping down on the foremost pedal.
To a lesser extent, you can also:


Lock out the front suspension, or dial it down a bit in bounciness.
Move your body weight forward on the saddle, or get off the saddle completely.
Remove weighty things from the rear of the bike - I found moving my full water bottle to the forward bottle cage had a noticeable impact.  Consider moving toolkits etc.