There Are Isolated Data Points, But There Are No Good Comprehensive Compilations, Only Estimates
There is no official source that calculates it, although public records exist in each court system with its own data base (one in some states, several in other states, plus the federal system) from which someone diligent could compile this data. There are no U.S. jurisdictions of which I am aware that report that number in that form on a statistical basis.
Almost every court system does publish annual reports that disclose how many cases of a particular types are filed and there are detailed statistics maintained by a private research group of detailed court filing statistics in the 100 largest counties in the U.S.
There are also academic estimates (mostly based on surveys and statistical samples) in various papers, compiled often for tort reform debates, and many casualty insurance companies have in house estimates that they share through industry associations related to tort claims that are used to determine how much to charge for insurance policies. Often these studies do a statistical estimate of the average amount of various types of cases, use official data on the number of cases filed, and adjust for outlier very large dollar cases on a fairly ad hoc basis, or make reports based upon insurance claims made.
There are also many economic estimates of how much contractual debt is outstanding and is in default, maintained primarily by credit reporting agencies and government agencies and select economics departments at universities, but that doesn't use the source of demands in complaints that you suggest.
As a practical matter, it is impossible to get a comprehensive national number of the basis that you suggest because some states have rules of civil procedure that prohibit complaints in civil actions from containing a demand for a sum certain of money damages in certain kinds of cases or in all cases, and including all of that detail in the body of the allegations of the complaint is common in some jurisdictions and largely deferred to post-complaint disclosures and discovery in others.
Even when a fixed dollar money demand is allowed, it is often not helpful because often a litigant will make a claim in "an amount to be determined at trial", especially in complex cases or where non-economic damages like pain and suffering are involved.
There are better compilations of statistics (although still only by private parties) of jury verdict amounts and also separately, of judgments entered in cases, in specific subtypes of cases, than there are of claims made. Tax authorities publish similar statistics.
There are also very big dollar definitional issues.
For example, do you count non-judicial foreclosures of real estate? If you do, do you include the full amount owed on the debt or only the deficiency judgment amount claimed in excess of the value of the collateral?
In some jurisdictions, tax debts are enforced through a separate tax collection authority process outside the courts, and in others, there must be a filing similar to a civil complaint in an ordinary civil court first. Whether or not you count those profoundly influences the total.
Do you count claims filed in bankruptcy and probate and receivership cases which are resolved in the first instance by a fiduciary and only go to a court if the initial resolution is disputed, or do you only count disputed claims, or do you not count cases like that at all? Do you adjust the filed claim amount total if the debts are discharged in bankruptcy or a similar insolvency process or probate claim process?
How do you count multiple overlapping claims for the same injury on multiple legal theories, for example, breach of contract and fraud, possibly with different dollar amounts of damages awardable for the same injury depending upon the theory relied upon?
How do you count claims of parties in equitable cases like boundary disputes or divorces involve property divisions, child support and alimony, or restraining order cases where some or all relief is "in kind"?
Do you include court costs and attorneys' fees sought, which are indeterminate when a case is filed, or only the principle amount? Do you include pre-judgment or post-judgment interest on a claim or neither or both?
What Do You Want?
To get the best answer it helps to know why you want to know. For some purposes, incomplete data is sufficient or even preferable.
Are you really interested in all cases, or only certain kinds of cases? There are many good studies of specific types of cases in the academic literature (e.g. physical injury and physical property damage torts, intellectual property lawsuits, class action lawsuits), but I'm not aware of any good comprehensive statistics, in part, for the definitional reasons identified above.