Why is 到 necessary here?
Combination of a verb with 到 generally implies some sort of "finished" state, in an sense it's similar to some sort of perfect tense:


我 走到 公園 / I walked to the park
我 吃到 炸雞了 / I finally ate that fried chicken
在字典裡 查到 這個字了 I found this word in a dictionary


Without the suffix "到", the verb along could be "unfinished"


我 走去 公園 / I walk to the park
我 吃 炸雞了 / I ate fried chicken
我在字典裡 查 這個字了 I looked this word up in a dictionary


Now back to your example:


  今天的考试要用到词典，请大家把词典也放在桌子上


Notice that "今天的考试" is not the subject of the verb "用到" -- since the "exam" would not be the user of a dictionary. The user of that dictionary is "大家". Alternatively that sentence can be rephrased to be: 


  今天的考试，大家要用到词典，请把词典也放在桌子上


I'd translated this to


  Each one of you will be using a dictionary in the exam today, please also put it on the table.


Now if we remove "到" 


  今天的考试，大家要用词典，请把词典也放在桌子上


IMHO it can be translated into something like any of these:


Everyone intends to use a dictionary...(omit)
Everyone is asked to use a dictionary...(omit)
Everyone must use a dictionary...(omit)


In a sense, adding "到" disambiguates from these different contexts -- well, usually appending more modifiers disambiguates better... :)

Now that was a different example then whan you provided, and I admit that it is a bit of a streatch. Let's revisit this one without the "到":


  今天的考試要用词典


If I must not insert an auxiliary subject into the sentence for translating, it is almost tempting to come up with something like:


The exam today will use a dictionary
Today's exam will use a dictionary


(Translating word-by-word. A pretty common thing being done by many English learners)

Overall the call-to-action is not differ and pretty clear -- dictionary on the table! -- but that's because the options are pretty limited.