Although we can't identify the orange mushrooms to exact species, they do look like a common type of saprophytic (only consuming organic matter that is already dead) fungus that might be risky to ingest (for people or pets), though we cannot speak to how hazardous they may be for either. We can't identify the white mushrooms from the description alone, and accurate mushroom ID even with photos can be difficult or impossible since so many look alike.

There is no treatment you can apply that would get rid of them, and they're actually a good component of a healthy ecosystem in the lawn, returning nutrients to the plants and boosting soil health.

If you irrigate the lawn regularly, make sure it's only being watered when the soil in the lawn is fairly dry to the touch about six inches deep. Watering more often will not only pose problems for the turf (and wastes water) but can support the growth of fungi that rely on moisture to spread via mushrooms. Heavy dew might have the same effect, and granted, some areas have been getting lots of rain recently, which of course you can't do much about. As the soil surface dries the mushrooms should go away.

All we can say is that, given the potential hazard for ingestion of the fungi, the dogs should be kept out of the yard when these mushrooms are present. Fortunately, individual mushrooms don't last very long and generally wither within a few days. (Make sure they're not eating grass that was treated with any other pesticide, such as grub killer, weed killer, a fungicide, or any other pest or disease control anytime that year.)

Miri

We will have to consult with our plant pathologist on Monday as to what family or genus of fungus he thinks each of these mushrooms belong to. He might not be able to get to genus level if they look too similar to unrelated species, though, as can happen often with mushrooms. Hundreds of mushroom-producing fungus species occur in Maryland.

If muzzles would keep the dogs from eating the grass (and therefore potentially mushrooms along with it), then perhaps so, but we defer to the advice of a vet in this case. If spores from the mushrooms wound-up on paws/fur that they later groom off and ingest indoors, we do not know how likely they are to pose a hazard to their health; it may depend on dog breed, and we have no training for veterinary concerns. If we can make an educated guess as to mushroom type, then you can ask a vet about the relative health risk for each.

Miri

Thank you for your patience; we had to consult with a second specialist to make an educated guess.

The orange mushrooms might be Marasmius siccus. There is a small gallery of additional images on Maryland Biodiversity Project's page, where they list a common name of Orange Pinwheel Mushroom.

The larger, pale mushrooms might be a species in the genus Mycena, but we don't have a more concrete ID.

Miri