According to the official Tempur website and their guidelines;
Yes you can use a Tempur mattress on a slatted bed.
You can also use a Tempur mattress on bed base that has a solid platform.
The only bed base that you cannot use a Tempur mattress with is an old fashioned spring type base.
Images created by UK.Tempur.com 
UK.Tempur.com
But…
I would disagree.
From experience, a platform top bed base is much better than a slatted bed base when using a Tempur mattress.
We used to have a rather nice solid wood frame bed.
Here is our old bed, (we had just set it up after moving house.) -There is our Tempur mattress just to the left of the bed.
We were totally happy with this bed and had used the Tempur mattress on it for about a year.
We originally bought this slatted wooden frame bed to use with a traditional sprung mattress.
(However, that mattress had to go, as it was contributing to us getting a bad night of sleep and regular back pain.)
Why we switched to a divan bed base for our Tempur mattress
That was until we were feeling at night like the Tempur mattress has suddenly stopped being as cushiony and supportive. It was as if the mattress has become squashed or compressed and was not giving full comfort any more.
This was when we delved underneath the bed, only to realise that our super squishy marshmallow bed was kind of seeping through the gaps in the bed slats like sausage meat!
Gross, I know. It seems that the weight of the mattress, even more so when we were lying on top of the mattress was forcing the soft base of the Tempur mattress down into the gaps between the slats.
Now, we are average sized adults, so weight definitely wasn’t an issue.
I think this uneven base was having a big impact of the comfort of the matter (even though it it supposed to be fine to use a Tempur mattress on a slat bed.)
The sinking weightless feeling that you are supposed to have when lying on the bed sort of disappeared slowly. And when we lifted the mattress up off the slats, there were grooves and ridges in the base of the mattress from where the slats had been pushing into the base.
We switched to a divan base bed
We have now switched to a flat platform style base, one with a solid base for the Tempur mattress to lie on and the mattress feels back to its old self. Extremely comfortable and very supportive.
If anyone was asking which type of bed to go for, to use with a memory foam mattress, I would always say go for a platform style divan bed base. To give your memory foam mattress more support and to protect it for its full life of use.
I hope this article was useful to you. If you have had a similar experience or have a question on this matter, leave a comment below.