When my washing machine(in my basement) is running, my bathtub(on first floor) burps and the toilet(also on the first floor) flushes and splashes and just carries on. The bathroom was recently renovated and now has a dedicated vent. This also affects the kitchen sink as well. This was a problem before renovation and we've checked every pipe for a blockage and no signs of a blockage anywhere.
Why is this happening and how can this be resolved?

You either have a plugged vent/vents, lack of vents, or incorrectly vented fixtures. Definitely sounds like a vent issue. Are there any other fixtures in the basemenst, and are they also making noises? What does your washer drain hose drain into, and is it a sealed connection, or is the hose just dropped into a standpipe or laundry sink like it should be.

I have a different take on this then MGD, To me a blocked vent will produce a "gurgle" as the discharge attempts to vent itself through a fixture trap or the toilet while a "burp" or "bubbles" indicate a back flow of air. Let me explain.
I think you either have a partial blocked or else the system pipes are too small to handle the high volatility discharge the newer washers put out, especially the newer Maytags.
Here is what I think is happening, Your washer discharges with much force and volume.
The discharge goes into the drainage system and either hits a partial or a pipe too small to handle the volume and bounces back sending a bubble of air ahead of it. If you have a older home with 1 1/2" galvanized pipe for the drainage you have found the culprit but if that's not the case I would snake out the washer line. Good luck, Tom

I agree Tom. However If the washer hose is in a standpipe, and not sealed (in the basement) , then I would think the water would just back up and overflow from the washer standpipe before making the upper level (first floor) fixtures do what they are doing. If the standpipe isn't sealed, I was thinking vent issues. If the standpipe is sealed, I totally agree that it may be a partially clogged house drain. Just seems like the water should be coming up the standpipe for the washer in the basement. Just something I have in my head I guess. Lets see what the OP has to say. Ill be gone for the weekend, and back asap. Take care and have a good weekend all.

Hey Dad,
Are talking apples and oranges here? You're talking about a over flow and I'm referring to the air produced by a discharge hitting as partial clog or attempting to get through a pipe that's too small to handle the volume. A blocked vent will pull air in as it attempts to vent while a blockage will bounce back pushing air ahead of it, And that's what I'm picking up on here. And your thoughts? Regards, Tom