Will These Burn Marks from LHR Heal Without Scarring? (photo)

Depending on the depth of the burn, there is a risk of residual hyper or hypopigmentation with healing.  The good news is that most of the time, superficial burns (and yours appears to be superficial) will resolve without any long-term pigmentary change.

It is extremely important to keep the area moisturized with a light, non-oily moisturizer and keep it absolutely out of the sun while it is healing.  Aloe tends to dry the skin.  Neosporin can potentially promote an allergic contact dermatitis, which would further inflame the area and result in more pigmentary change.

Do not rub or scrub the area in an attempt to remove the crusts.  Let them fall off on their own.

As a preventative measure, you have to make sure that your skin has not been exposed to the sun for at least 6 weeks before and after the laser procedure as that changes the amount of melanin in the epidermis (top layer of the skin) and makes the difference between skin color and hair color smaller.  That would make the superficial pigment more likely to be affected by the laser procedure.
 

It is difficult to tell from your photo whether or not the spots are red or brown.  If they are red, they might be superficial bruises and will resolve shortly.  If they are brown, they will most likely fade with time, but please use sunscreen at all times if the areas will be exposed to the sun.  You can also try a bleaching cream if the spots persist for more than a month after your treatment.

My recommendations for treatment of burns in the skin are as follows:
1. Keep the areas clean by washing with a mild soap twice daily.
2. If there is blistering, relieve the pressure and hasten healing time by gently popping all blisters using a sterile needle or pin at the side of the blister. This should be a painless procedure. Do not remove the roof of the blister but allow it to settle down so it is flush with the skin. The blister roof will then form a protective cover.
3. Apply Aquaphor Healing Ointment (available without a prescription) to all involved areas to keep them "greasy". The greasier the area is, the faster it will heal. The drier and scabbier the area gets, the slower it will heal.
4. Permanent scarring is unlikely but you have to guard against post-inflammatory hyperpigmentation by constant protection from the sun for these areas such as using protective clothing and sunscreens.
5. Make certain you are under a doctor's care and watch for signs of infection so that oral antibiotics can be started as soon as possible if necessary.