This was originally going to be a single-part topic, but I realized after several comments on the original post, that I needed to clarify and address a few extra things for both the unbeliever and the believer.
First, I want to define a term. Second, I will discuss the scriptures brought up as challenges. Third, I will provide scriptures that allow for "doubt" being biblical.
1. Doubt- when I use this word in the original post, I am referring only to be skeptical of the truth of a claim. I'm not talking about distrusting someone, and I am not talking about not believing that something you pray for will come true. I am speaking strictly of being unsure of the truth of something someone is claiming to be true.
2. A few scriptures were posed to me as being Biblical evidence that doubting truth claims is unbiblical. The implied conclusion seemed to be that investigating Christian truth claims is discouraged in the Bible.
Here are the scriptures posed as challenges:
Matthew 21:21 (parallel passage Mark 11:23)
Let's look at each one individually:
"If any of you lacks wisdom, he should ask God, who gives generously to all without finding fault, and it will be given to him. But when he asks, he must believe and not doubt, because he who doubts is like a wave of the sea, blown and tossed by the wind. That man should not think he will receive anything from the Lord; he is a double-minded man, unstable in all he does."
Notice that the context of this passage is a man asking God for something (wisdom, in this case). If a man asks God for wisdom, but does not believe (doubts) that God will give it to him, he is "like a wave of the sea...". This passage's specific definition for the word "doubt" is "doubting someone" or "not believing that something you pray for will come true." Neither of which is the definition or context of my original post. Since the context of the two being compared (my post and the biblical passage) is not the same, the comparison is unwarranted, and may be discarded.
"Jesus replied, "I tell you the truth, if you have faith and do not doubt, not only can you do what was done to the fig tree, but also you can say to this mountain, 'Go, throw yourself into the sea,' and it will be done."
This one has also a different context and definition. (I have always been a fan of Greg Koukl's booklet and online article "Never Read a Bible Verse". The short version is that one should never take only a single verse (out of context) to make a point.) If you read the whole story (Matthew 21:18-21), you will notice that Jesus is not talking about doubting the truth of something that is being told to you. He's talking about being able to wilt a fig tree or move a mountain. The specific definition of "doubt" and overall context is not the same. So, that really has nothing to do with my post either.
"But the man who has doubts is condemned if he eats, because his eating is not from faith; and everything that does not come from faith is sin."
Please read this one in context (Romans 14- the entire chapter). Notice that Paul is talking about trivial issues that may cause someone to stumble (food and drink in this case). He makes it clear in the verse 23 (the one you quoted) that he is still talking about food and drink. Specifically, Paul is talking about doubting (not knowing) whether or not eating or drinking something will cause a brother to stumble. Once again, this is not the definition of "doubt" nor the context that I use. So, this one is not applicable either.
3. Here are a few scriptures that promote testing truth claims, being gracious to (rather than condemning) those who have questions (doubt), and providing a defense of the Gospel.
1 Peter 3:15
1 Thessalonians 5:21
1 Peter 3:15
But in your hearts set apart Christ as Lord. Always be prepared to give an answer to everyone who asks you to give the reason for the hope that you have. But do this with gentleness and respect
In 1 Peter 3:8-22 Peter is commanding his brothers to be compassionate and humble in everything that they do. In verse 15, he specifically mentions situations in which someone is asking questions about their beliefs.
1 Thessalonians 5:21
Test everything. Hold on to the good.
In 1 Thessalonians 5:12-28 Paul is saying pretty much the same as Peter above.
Be merciful to those who doubt; snatch others from the fire and save them; to others show mercy, mixed with fear—hating even the clothing stained by corrupted flesh.
In context (Jude 1:17-23) Jude is speaking of people who doubt the truth-claims of Christianity.
I have published two other posts that may also help to shed more light on the subject (each has references for further investigation):
Is "Blind Faith" Biblical?
If you think that I need to address anything else in regards to this topic, please let me know.