Lots of people view the birth of a baby as a good reason to get rid of their 'pet'.  This is an unfortunate view point and the worst mistake most people will ever make when parenting.  Millions of animals die alone in shelters everyday...a big factor in this (from my experience) is the new baby situation.  Being in rescue I also get emails on an almost daily basis asking me if I can take their beloved cat/dog/other because they are suddenly pregnant.  People in this situation tend to believe that this is acceptable behavior and are often proud that they contact a rescue to take the animal(s).  What they don't realize is that these animals have feelings and the rescue/shelter workers are left to pick up the pieces of a broken home.
I have one child who is 4 years old, am having a new baby tomorrow, 6 dogs, and 2 cats.  Would I ever think of abandoning an animal to 'make room' for the baby?  Never.  Will there be an adjustment and transition phase?  Absolutely.  I am ready and willing to go through it with my human and animal family members.
A request I often get goes something like this "I already have one (or more) child and one on the way.  I know how draining and time consuming it is to have a new baby and I am afraid that I will not have time for 'Fido' anymore.  He loves my child and is so good with him/her.  We really want to have 'Fido' go to a good home."  First of all what are you teaching your child by abandoning one family member for another?  That it is acceptable to disregard one life for another?   How will you explain it to your child that you gave away his companion?  'Fido' has been good to your child so why wouldn't 'Fido' be good to another child?   So then there is the time factor.  Having been thru this once before I can say either animals or children are time consuming.  Having a new baby is a change, yes, but so is getting a hair cut.  It is ok to ask for help with your animals or your child.  I had help with both for a while.  My neighbor would watch Jake while he was sleeping so I could spend some time with Disco outside.  Then to return the favor I would go out and play with her dog too.  (Keep in mind I was a single mom at this time...if I had a partner in all this it would have been even easier.)  In the winter when I would go out for the animals I bought my baby a snow suit, put him in a 'sling' (like a front wearing backpack) and zipped up my coat around him.  For rain I bought an oversized poncho that would fit over us.  When he got too big for the sling I got a stroller.  Problem solved.  Babies sleep the majority of their time allowing you time to take a nap, play, anything with your animal(s).  
Another request I get to take an animal is that the animal is not well behaved.  Who's fault is that?  Not the animals, its the persons.  First question I always ask "Is the animal spayed or neutered?".  Most of the time the answer is no and reasons usually are that the people could not afford it.  So then I ask if the animal has seen a vet and that answer is no most of the time as well.  Well...in my opinion how are you going to pay for the care of a child of you can't pay for the care of an animal?  So I try to direct them to low cost/no cost vet programs in their area.  I have even offered to take the animal round trip.  Some people do not understand that spay and neuter can drastically change an animals behavior for the better.  Not only are you saving lives by doing it but you are making the animal healthier by preventing a host of health problems. 
I also ask what the animals living arrangements are.  A happy, well adjusted dog does not come from living his/her life on a chain alone in the yard and the same goes for a cat that lives its life in the closet.  Keeping an animal in the house is ideal however you can't keep the animal penned up for 10 hours a day and expect him/her to come out of confinement and not be excited.  A good way to look at this is a grade school age child and recess.  Recess was created to let the child 'get it out of their system' so they could return to school and behave.  Animals need the same.  However animals, like children, need human companionship too.  I know that you can call a high school and hire a student to come over and play with your animals.  You can also call a professional dog sitter or rescue.  Remember that well socialized animals make the best companions.  If this means you need to seek help from a trainer to get them to behave then do it.  If your child has Attention Deficit Hyperactivity Disorder would you give them away?  (Some people believe think this is not a good analogy, however when you commit to an animal companion it is the same as adopting a new member of your family.)  Many rescues and shelters can also direct you to someone that can train you to train your animal.  I know from my end I will do everything I can to help you keep the animal in your home and most people in this industry will do the same.
The last point I want to bring up is the effect on the animal.  Any species will know that they are now unwanted.  Much like a child suddenly dumped into foster care they will wonder what they did wrong, have anxiety, guilt, depression, and attachment issues.  This usually cause behavioral problems above and beyond what they might have had in the previous home.  These animals have a hard time trusting people or other animals, forget their training (assuming they had any), and resort to behaviors that are just to get attention (chewing, fighting, hiding, using the house as a restroom) no mater if it is positive or negative.  I have had several sofas, love seats, chairs, tables, beds, shoes, clothes, etc destroyed by animals.  It can happen fast too...like when you are in the bathroom.  It does make me angry but not at the animal...its at the person.  So who is left to pick up the pieces?  Not the dumper.  They get to go on with their lives and eventually the memories of the animal will fade.  I, however, get to make this being whole again. 
I always end the conversation with "go sit down with your animal.  Look into their eyes.  What do you see?  Love, devotion, caring?  Now go tell your animals you don't want them anymore.  Go have every member of your house tell them that they (the animal) are not welcome in their lives anymore.  But you wish the best for them.  What will your animal do?  They will not bite you.  They will not cry.  They will look at you with the love they always have and try to comfort you because they see your discomfort and can't stand to see you suffer." 
 Remember that if you are considering giving away your animals.