"THEM THAT BELIEVE" refers to ONLY the APOSTLES.

Reason 1:  Context is the Apostles' Unbelief in Christ's resurrection.
In Mark 16:14, Jesus rebukes the apostles for their unbelief in His resurrection.
In v. 11-14, four times the apostles' unbelief is mentioned as Jesus addressed the apostles:
v.11 "They (apostles), when they had heard that he was alive ......believed not".
v.13 "...neither believed they them".
v.14 "He...upbraided them (apostles) with their unbelief and hardness of heart".
v.14 "...because they (apostles) believed not them which had seen him after he was risen".

After this rebuke Jesus gives the apostles the Great Commission in v.15,16.  Once the commission is given, Jesus returns to his immediate audience of the eleven apostles when He says in v.17,18 "These signs shall follow them that believe" (ie. those of you eleven apostles who believe in my resurrection).

In my name:
1.  they shall cast out devils.
2.  they shall speak with new tongues.
3.  they shall take up serpents.
4.  if they drink any deadly thing, it shall not hurt them.
5.  they shall lay hands on the sick and they shall recover.

Reason 2:  Practical Experience shows that promises 3, 4, 5, are not true for every believer today. In spite of continued failure to heal Charismatics still think that these 5 promises are for every believer today. 
The obvious problem with this interpretation is that it does not deal with everything mentioned in the text.

a)  No Charismatic can always fulfill the 5th promise.  It says that they shall lay hands on the sick and they shall recover.  This means 100% recovery all the time.  No one can do this today, as the apostles could in Acts 5:12-16.
 "By the hands of the apostles were many signs and wonders wrought among the people...they (believers) brought forth the sick into the streets... and they were healed every one".  v. 16.
John Wimber, a famous Charismatic, laid hands on 250 Down's Syndrome children and none were healed.
Reinhard Boncke tried to heal over 200 Kenyans and none were healed at a meeting at Kisumu.

b)  "If they drink any deadly thing, it shall not hurt them".
"If" means accidentally.  The problem is that many good Christians have become sick or died by poisoning.  Have you as a believer ever vomited?

c)  "They shall take up serpents".
We ask a Charismatic to take up a deadly snake.  He replies by saying: "No, that would be tempting God."  Yet in the first century this was a sign that the apostles did (shall do) to prove the Word of God and their authority.  Some believers today have died in trying this.  Hence showing that this is not for today.

Reason 3:  AORIST tense of "believe" in Mark 16:17 refers to those who did believe in the past, not who will believe in the future.
"These signs shall follow them that believe."  The Greek word for believe here "pisteusasin" which is in aorist participle referring to those who did believe in the past".
[S. Zodhiates, "Complete New Testament Word Study Dictionary" p. 1107]
"not those who would believe in the future".
[S. Zodhiates, "Hebrew-Greek Key Word Study Bible" p.1223]
The aorist tense refers to a past action at a point in time, not an event that recurs again into the future, as it would be if it was true of all believers.

Reason 4:  Belief (pisteusasin in Mark 16:17) does not refer to saving belief.  In John 3:16 and 5:24 saving belief is "pisteuwn", a different word ending.  Hence, "belief" in Mark 16:17 refers not to all future believers, but to the apostles believing in Christ's resurrection at a point in time, once for all time.

Reason 5:  "Do all speak with tongues?"  1 Corinthians 12:30, requires the answer "No, not all believers have the gift of speaking with tongues".

Reason 6:  The Purpose of these signs was to confirm the Word of God spoken and written by the apostles as Mark 16:20 says.
"And they (apostles) went forth and preached everywhere, the Lord working with them (apostles), and confirming the Word with signs (eg. tongues) following".
Once the Word of God (the New Testament) had been confirmed by the apostles' signs, then further signs were not needed.  The signs had done their job of proving the New Testament as being from God.  When the Word of God was completed, then miraculous Apostolic confirmatory signs were no longer needed.

Reason 7:  Mark 16:20 continues the context to mean apostles.

This describes what happened to the apostles after they changed from unbelief in Christ's resurrection to believing it and preaching it everywhere with signs following.
Hence in v.20, the context is still only the apostles doing signs, not every believer doing signs.

Conclusion.  Either all 5 signs are valid for today or none are valid today.  They only refer to first century apostles.

Reason 8:  Tongues had ceased by 70 AD because their use is only mentioned in the earlier New Testament books, such as Mark 16 (57 AD), Acts (54 AD), 1 Corinthians (57 AD).
In all the latter books, tongues are never mentioned.
Why?  Because they never happened after 70 AD and were of no relevance to any churches thereafter.  Tongues are never commanded in any New Testament book for believers to practice.
1 Corinthians 12, 13, 14 rebukes the Corinthians wrong use of tongues.  It is not endorsing tongues to be practiced.  Charismatics think that the key to spirituality is tongues, a phenomena that is not mentioned in 24 out of 27 New Testament books.

Why are tongues ABSENT from 2 Corinthians to Revelation?  If tongues were so necessary, you would definitely see them in the rest of the New Testament.
The book of Romans, which is well-known as the most complete summary of Christian doctrine and practice in the Bible, does not once mention tongues.  Why not?  Because tongues, by the time Romans was written (60 AD) was almost finished as a gift and would be of no relevance in the Church age after 70 AD when God judged Jerusalem.
Think about it, if tongues were important, God would have mentioned them at least once in Romans, or 2 Corinthians, or Galatians, or at least in one letter to a church in a positive context.

Reason 9:  The gift of tongues was never endorsed or practiced by early church leaders.  

Clement of Rome in 95 AD wrote a letter to the Corinthian church rebuking every problem that Paul rebuked except tongues.  Why did he not mention tongues or miracles? 
Justin Martyr (160 AD) visited many churches but never once mentions tongues, not even in his lists of spiritual gifts.
Origen (250 AD) in his voluminous writings never mentions tongues, but argues against Celsus that the sign gifts of the apostles' age were temporary and were not exercised by Christians in the 3rd Century.
Chrysostom (347-407 AD) commented on the 1 Corinthian passage:  "This whole place is very obscure:  but the obscurity is produced by our ignorance of the facts referred to and by their cessation, being such as then used to occur but now no longer take place."  [Homilies, XXIX, 1]
Augustine (354-430 AD) said in speaking of Acts 2:4, "In the earliest times, the Holy Ghost fell upon them that believed: and they spake with tongues...these were signs adapted to the time.  For there behooved to be that betokening of the Holy Spirit...that thing was done for a betokening and it passed away."

Reason 10: "Now abideth faith, hope, charity, these three..." 1 Corinthians 13:13.
Now (57AD) abides (continues) faith, hope, charity, these three" (not "these six prophecy, tongues, knowledge").
Since faith and hope finish at Christ's return, then prophecy, tongues and knowledge had to have finished by 96 AD because they were not abiding, but declining in 57 AD.