File them somewhere where you can find them when you renew your license. You'll be asked how many CEs you've earned, and some states make you provide the information on the papers. In CA you have to keep the information for a minimum of three years.
Keep your certificates of completion. When you renew your license if you are asked for the info about your CEUs then you must provide it. I believe it is four years in CA. I was audited on my CEUs two years ago and had to send in copies of my certificates to the Board. Had I not had the certificates on hand, I would have had to remember what I put down on the form and contact the issuing authority for copies. Hassle.
Hold onto them. If you are audited, and cannot find all your certificates of completion, (even though you KNOW you did them), you will have to take an additional number of CEU's, turn those in and pay a fine.
I found that out the hard way. Now I can lay my hands on several years's worth at any given time. I keep better track of those than my bank account.
Also, there is no such thing as "CEU", although it is a common mistake to use this term. The actual term is "contact hours". In the past continuing education units were 10 contact hours = 1 unit. Some states required a specific set of units (2 units= 20 contact hours). Finally ANCC decided to drop the term and use contact hours only. Other organizations followed suit. Just FYI.