Not necessarily, but ceteris paribus it would, although not exactly via the channel you mention.
Government Bonds Basics
First, to be more precise, government primarily raises revenue by issuing bonds. Bonds are fixed income securities and they do not really have interest rate but coupon rates. The coupon rates might be different from prevailing interest but because bonds do not necessarily sell at their face value (e.g. government that issues \$1000 bond might be able to sell it only at \$900) there will always be some implied interest rate on any bond called yield to maturity (YTM). Important thing to know here is that, ceteris paribus, if bond prices increase YTM decreases and vice versa if bond prices fall YTM increases. This is important because bellow I will be talking about government bond prices.
Determinants of Bond Prices
Prices of government (or really any) bonds, and thus YTM/interest on government (or again any) debt will depend on demand and supply. The higher demand for bonds (ceteris paribus) the higher bond price will be and thus smaller YTM and vice versa. The higher the supply for bonds (ceteris paribus) the lower the price will be and thus the yield/interest will be higher. More specifically the main demand and supply factors according to Mishkin et al Financial Markets and Institutions are:
Wealth - bonds are often used as a vehicles to store wealth across time. Ceteris paribus, as people get more wealthier they demand more bonds and the lower interest rate will be.
Expected returns on bonds relative to alternative assets - peoples demand for bonds depends on other alternatives. People can save their wealth in form of stocks, or other equities or lend money to firms instead of government or other governments. Ceteris paribus, because of competition, if a person can invest into equity that has the same risk and controlling for all other relevant factors, with return on that equity being 10%, well then no matter what coupon rate will government offer on the bond, the bond will never sell for a price other than the price that would imply 10% YTM/interest. Otherwise why would people ever buy the bond in the first place?
Expected return is also affected by inflation, because people care about real return not nominal return and the relationship between real and nominal return is affect by inflation via the fisher equation: $i \approx r + \pi$, where $i$ is nominal interest rate $r$ real one and $\pi$ is inflation (see Mankiw Macroeconomics pp 110). However, as discussed below inflation is not necessary result of government spending.
Risk of bonds relative to alternative assets - this is almost self explanatory, people need to be compensated for higher risk with higher YTM/interest
Liquidity of bonds relative to alternative assets - people can sometimes find themselves in financial distress and in need to convert their savings into cash. Bonds that are more liquid (easier to sell) will be able to command higher prices and thus lower YTM/interest (ceteris paribus)
When it comes to supply, in case of private bonds there are many factors (such as investment opportunities), but in case of public bonds it is mainly deficit spending (even though deficit spending itself might depend on various factors, in macro we often think of it as exogenously given).
If government supplies more bonds to the market, then ceteris paribus the price will be lower and thus the yield/interest higher. This is so because the amount of savings economy has is typically limited so as government wants to take more and more people's saving it has to offer higher interest rate as it competes with other firms and governments that want to borrow that saving. However, in a situation where we have savings glut it will take very large increases in supply of bonds to move prices significantly.
Government Spending and Inflation
Government spending by itself does not necessarily raise inflation. Inflation (in the long run) is monetary phenomenon (see discussion in Mankiw Macroeconomics pp 106). There are some caveats and asterisks to that statement, for more discussion see the above source, but government spending per se won't raise inflation if the government spending is not financed by new money (e.g. when central bank purchases bonds with newly created money through OMOs).
The reason for that is that even though it is true that (any not just government) spending is required for inflation to materialize, it is just a channel for effect of money supply. If money supply is constant, and if we for a second simplify and assume closed economy, the only way how government can borrow money and spend it is for some of its citizens to save and thus not spend their money but give it to the government by purchasing its bonds.
Government spending does not magically have more effect on aggregate prices than private spending. Consequently, unless people who lent money to the government would otherwise just put that money under their matrasses and not spend it (which can be true in some situations like recessions with liquidity traps), the increase government spending will be offset by decrease in private consumption and investment spending, so it would have no effect on inflation. Consequently, unless you add more specification to your problem (e.g. are you talking about massive government spending during recession or massive government spending financed by monetary expansion?) you should not a priori assume it increases inflation.
In conclusion, you cannot simply tell if interest rates will raise just because of large government spending financed by debt. Government debt financed spending does increase interest rate, but there are other factors in play. If government debt financed spending coincides with increase in wealth (which would put pressure on yields/interest to go down) nothing may change as the two effects might offset each other.