Should we put God to test?

If we define ‘tempting God’ as provoking him and to anger through unbelief, when our own experiences and God’s own manifestations of his power has clearly confirmed to us the certainty of his promises, the tension is dissolved.

Let’s show by example:

Luke 4:12 ... Do not put the Lord your God to the test. (In other words I am so satisfied with my father’s care and confident in his promise that I need not veer any way off it seeking another).

Judges 6:39 “Allow me one more test with the fleece, but this time make the fleece dry and let the ground be covered with dew.” (In other words God’s will was not yet so fully clear and do to the weakness of his faith Gideon asks for further proof which God allows and does not consider it a provocation but like the man who humbly said ‘I believe but help me with my unbelief’. Mark 9:24).

Matthew 12:39 A wicked and adulterous generation asks for a miraculous sign! (Clearly only a wicked generation would not recognize the abundant proofs already shown by Christ and greatly provoked God to anger in resisting it!)

2 Kings 20:8 Meanwhile, Hezekiah had said to Isaiah, "What sign will the LORD give to prove that he will heal me and that I will go to the Temple of the LORD three days from now?" (The King Hezekiah is in a similar situation as Gideon in Judges 6:39 like the man who said ‘I believe but help me with my unbelief’)

Deuteronomy 6:16 You shall not test the LORD your God, as you tested him in Massah. (This exactly the evil provocation that the Jews in the Jews in the desert provoked God through their unbelief after so many miraculous proofs, that truly defines the wickedness of this ‘tempting’)

Genesis 15:8 But Abram said, "O Sovereign LORD, how can I know that I will gain possession of it? (Again this is a humble request of a man filled with faith, rather than being a provoking of the Lord, Abraham is known for such a great faith he had with such little evidence made to him!)

So we see there are no conflicts though admittedly on first glance it does appear to conflict and is a god question.

In a manner of speaking, there are two kinds of tests. These two kinds of tests correspond to two kinds of doubt: the "I don't want to believe" doubt and the "I do want to believe" doubt.

Unbelieving doubt

This is where someone doubts a statement or a miracle because they don't want to believe it. Prime examples are the multiple times where Jews doubted the miracles of Jesus and repeatedly asked Him to show them a sign. In fact, there is this exchange:

John 10:24-25 (NIV)

²⁴ The Jews who were there gathered around him, saying, “How long will you keep us in suspense? If you are the Messiah, tell us plainly.”

²⁵ Jesus answered, “I did tell you, but you do not believe. The works I do in my Father’s name testify about me,

Here, the Jews were called out as completely disregarding all the miracles that Jesus did up to that point, including healing a man blind from birth, which no one had ever done before (John 9:32). There are many other examples in the Old Testament of the Jews scorning the signs God had already shown them, thus angering God and bringing punishment upon themselves. These people doubt because they don't want to believe.

Believing doubt

This kind of doubt is more along the lines of "I want to believe, but I don't think I can/should". The classic example is the demon-possessed boy's father:

Mark 9:21-24 (NLT)

²¹ “How long has this been happening?” Jesus asked the boy’s father.

He replied, “Since he was a little boy. ²² The spirit often throws him into the fire or into water, trying to kill him. Have mercy on us and help us, if you can.”

²³ “What do you mean, ‘If I can’?” Jesus asked. “Anything is possible if a person believes.”

²⁴ The father instantly cried out, “I do believe, but help me overcome my unbelief!”

Very clearly, here is the desire to believe.

There is also occasionally the need to doubt, as in the case of false prophecies, spirits and messiahs:

1 Thessalonians 5:19-22 (NLT)
¹⁹ Do not stifle the Holy Spirit. ²⁰ Do not scoff at prophecies, ²¹ but test everything that is said. Hold on to what is good. ²² Stay away from every kind of evil.

1 Timothy 4:1 (NLT)
4 Now the Holy Spirit tells us clearly that in the last times some will turn away from the true faith; they will follow deceptive spirits and teachings that come from demons.

Matthew 24:24 (NLT)
²⁴ For false messiahs and false prophets will rise up and perform great signs and wonders so as to deceive, if possible, even God’s chosen ones.

And in order to identify these falsities, one must test them...

Conclusion

What's important here is not the existence of doubt, but the attitude associated with it.

1 Samuel 16:7 (NLT)

⁷ But the Lord said to Samuel, “Don’t judge by his appearance or height, for I have rejected him. The Lord doesn’t see things the way you see them. People judge by outward appearance, but the Lord looks at the heart.”

Testing the Lord is not what matters; what matters is why you're testing the Lord. As a final note, there is one place in the Bible where God actually tells us to test Him:

Malachi 3:10 (NIV)

¹⁰ Bring the whole tithe into the storehouse, that there may be food in my house. Test me in this,” says the Lord Almighty, “and see if I will not throw open the floodgates of heaven and pour out so much blessing that there will not be room enough to store it.

The answer is simple and these fragments all support it:

No - we shouldn't test God. We shouldn't demand revelations, we shouldn't demand signs and miracles, as it is stated in the fragments you are citing.

Yes - we should test revelations. As Saint John the Teologian states:

Beloved, believe not every spirit, but try the spirits whether they are of God (1 John, 4, 1)

All these people you see in Bible "testing God" received some revelation, but were not sure if it is from God, or if they are in some kind of delusion. They don't doubt there is God. They doubt if the revelation is genuine. That's why they put it to the test.