This question can be difficult for some to answer because with the disadvantages, you have to consider the advantages as well. For example, while immigrants may utilize resources that current residents are using, without those immigrants, we may not have the resources we have. For example, while immigrants may utilize our medical resources that keep the "original citizens" healthy, it is fact that without those immigrants, we may not have the medical resources we have today. For instance, a lot of those immigrants become staff personnel in those medical offices, aka doctors and nurses, who keep us healthy.
Also, if we are having a problem with our natural resources because of immigration, rather than just limiting immigration, many times, those immigrants may have better answers to those problems as well as many problems we have here already.
As an aside that is personal opinion and only indirectly related to your question about disadvantages of immigration, our current situation with illegal immigration and resource utilization comes to mind. Even with the illegal immigration occurring right now, are they utilizing much of our resources right now? Yes. But, deport them all? That would take even more of our natural resources, as in money to simply fund the operation, increasing the number of immigration agents to handle the operation, and in general making government bigger. But, if we provide a pathway for citizenship for the illegal immigrants, on that pathway, we could require things like "paying a fine," which becomes income for our country. Once they become citizens, they would become taxpayers to this country, which income for us. Are some criminals? Sure, just like some current citizens are criminals. In short, there's not that much difference between them and us when we first came to this country hundreds of years ago. We were illegal immigrants then, just looking for a better way of life. So are they.
Immigration refers to the influx of foreign citizens into a country or region. People immigrate, typically, in search of better employment opportunities, better educational and learning facilities or security. While it may be beneficial to the local economy and (certainly) the immigrants, this may disadvantageous to the residents of the country (to which immigration takes place).
Immigrants utilize resources of the country, which gets divided between original residents and immigrants, thereby reducing the per capita availability to the original residents. Many immigrants also compete for the jobs, thus reducing the chances of original residents. And immigrants may also be willing to do a job at lesser fee, thereby decreasing the average labor cost in the market (more supply, less or same demand).
No doubt that many of these immigrants add value to the country and economy, yet most of them still have strong ties to their country of birth, and this results in efflux of wealth to those countries, in the form of gifts and remittance. All this may be disadvantageous to local residents (whose businesses immigrants may compete with) and to the country's economy.