It is normal. The sum of residuals is zero only if the intercept is fitted too. If intercept is fixed to 0 the sum of residuals is different from zero. If your value of intercept is not significantly different from 0 simply say that you do not have a probabilistic support to say that your intercept is different from 0, not that the intercept is zero

It is rare that a linear regression without an intercept should be conducted. Keep the intercept in your model, and don't worry that it wasn't significant.

Sal Mangiafico is perfectly right. !

Yes, for what is it important that the intercept is significant anyway? If you would center the predictors for example, intercept would change, but the rest would remain (basically) the same.

there's a bunch of similar posts on the internet and this same forum.

At least google before your start a new thread (that is what the guidelines of RG tell you anyway, along with common sense).

From what you say you remove the intercept because it not significant, which is known as bad practice: you should include regressors in your model according to theory or design regardless of their pvalues.

Thanks, Marco Acutis . I know that's what you said in your reply; I was just condensing it for clarity.

Actually, you should not use an intercept term if you know that when all independent variables (xs) are zero, y will be zero. The simplest case is a ratio estimator, and there, yes the estimated residuals do add to exactly zero every time, not just in the expectation.  Please see https://www.researchgate.net/publication/261474011_The_Classical_Ratio_Estimator_Model-Based.  (Note that a model-based classical ratio estimator is for a weighted least squares regression.) 

Also, please see pages 231 and 232 in Särndal, C.-E., Swensson, B., and Wretman, J.(1992), Model Assisted Survey Sampling, Springer-Verlang.  There they give conditions for this in general. 

When people say you must have an intercept for this unbiased situation, they are implicitly stating that the model is homoscedastic (OLS).  This is not the natural state of things.  See https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity, as determined by Ken Brewer, and Brewer, K.R.W.(2002), Combined Survey Sampling Inference: Weighing Basu's Elephants, Arnold: London and Oxford University Press, especially mid-page 111.  (Ignore the last paragraph on that page, there appears to be an editorial error confusing two things in the book.) 

Further, towards the top of page 111 in Brewer(2002), we see this:  "...a zero mean for the estimated residuals can only be achieved by way of including an intercept term if the model used reflects an assumption that the residuals themselves are homoscedastic."  But he goes on to say that assuming homoscedasticity does not actually make residuals homoscedastic.

Thus, we have well-behaved sum of estimated residuals for OLS only if there is an intercept term, but there may be heteroscedasticity, which is a feature to be expected, not a bug.  Often heteroscedasticity is present, but it is not always obvious.  Please see https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression, and various updates. 

With heteroscedasticity, you can achieve model-unbiasedness, with a zero intercept. 

So ...

If you impose homoscedastic, then you are stuck with an intercept term to get unbiasedness, even when that intercept is not actually different from zero.

Cheers. 

PS - 

So ... what to do:

It sounds like you need to use weighted least squares (WLS) regression. Use the examples in https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity to see how to estimate the coefficient of heteroscedasticity.  You can use https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx to apply this to your data. (There are notes on a sheet there about defaults when sample sizes are small.)

First, in the above I said "The simplest case is a ratio estimator, and there, yes the estimated residuals do add to exactly zero every time, not just in the expectation." Actually, that is for a particular ratio estimator, the classical ratio estimator, CRE, which is for a particular weighted least squares (WLS) regression.

The reference to pages 231 and 232 in Särndal, Swensson, and Wretman(1992) that I gave above is more complete, but for a quick, easily understood proof by Ken Brewer for the case of one regressor, showing that with no intercept term you get not just model-unbiasedness in expectation, but exactly, with a certain coefficient of heteroscedasticity, gamma, see

https://www.researchgate.net/publication/261474011_The_Classical_Ratio_Estimator_Model-Based,

pages 1 and 6, with more on pages 11 and 12. Information from pages 1 and 6 is attached. (I found this special case with gamma=0.5, the classical ratio estimator, to often be very useful for official energy statistics, from establishment surveys, in frequently applied production/data publication.)

Niladri  - 

To be clear, you are saying you don't think you need an intercept term as it seems very small at most.  Right?  It could easily be either negative or positive?  (Perhaps, from a subject matter perspective, you'd expect y to be exactly zero when all x's are exactly zero?  Is that true?)  But without an intercept term, you lose model-unbiasedness, i.e., the sum of the estimated residuals becomes more than trivially different from zero.  Is that correct?

You asked "Can you suggest me what should I do?"

I suspect you were using OLS regression, and you need to use weighted least squares (WLS) regression.  

In https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression, under the update, "Model-Unbiasedness: Another Problem with Implicitly Assuming Homoscedasticity," I have "...that when people say that an intercept is needed to have model-unbiasedness, they are implicitly 

assuming OLS (homoscedastic) 

r​e​g​r​e​s​s​i​o​n​.​" 

In a quote from Ken Brewer in an earlier response above: "Further, towards the top of page 111 in Brewer(2002), we see this:  '...a zero mean for the estimated residuals can only be achieved by way of including an intercept term if the model used reflects an assumption that the residuals themselves are homoscedastic.'" 

An exact zero mean is not required for model-unbiasedness, just one in expectation, and models can be complex, and the degree of heteroscedasticity varies (as measured by the coefficient of heteroscedasticity, gamma, in URLs noted previously), but I think your situation may be clear: 

My guess is that you are probably using OLS and need to use WLS regression. 

Cheers  - Jim

James R Knaub I'm saying that my intercept is not coming significant in my model using t test. So I didn't consider intercept in the model

Niladri Chakraborty -

Could we see your model?

If all predictors are zero, would you expect y to be zero? Then, no intercept term should be used.

Is this model heteroscedastic? (Essential heteroscedasticity is a feature, not a bug. It is to be expected that larger predicted-y-values be associated with larger sigma of Y, which is approximated by larger sigma of estimated residuals.) A homoscedastic model, which is often somewhat artificial, requires an intercept thrown in to make estimated residuals sum to zero. Perhaps you can avoid that.

It would be good to see a "graphical residual analysis" for your sample. (A "cross-validation" may be needed to avoid too closely fitting to your particular sample.)

"Significance" is often misleading.

Cheers - Jim